GCSE Maths: Equations from geometry: perimeters and angles (Foundation)
Part of the GCSE Maths linear equations chapter (Foundation). Chapter overview · Higher version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Equations from geometry: perimeters and angles
Diagrams often give lengths or angles as expressions with x in them. To find x we need a fact that tells us what the lengths or angles add up to. This gives us an equation. Here are the facts we will use (you meet them again in Chapters 17 and 18):
- The perimeter of a shape is the total distance around its edges: add up all the sides.
- Opposite sides of a rectangle are equal. All the sides of a square or rhombus are equal. Two sides of an isosceles triangle are equal.
- The angles in a triangle add up to 180°. The angles on a straight line add up to 180°.
- The angles round a point add up to 360°. The angles in a quadrilateral add up to 360°.
- The two base angles of an isosceles triangle are equal.
The method is: (1) say which fact you are using, (2) write an equation, (3) solve it, (4) put x back into the expressions on the diagram to find the length or angle that the question asks for, and (5) check, for example by adding the angles to see that they total 180°.
Diagrams in this book are drawn to scale, but in an exam a diagram may say “not drawn accurately”. Do not measure: use the facts.
Key ideas
- Perimeter = the sum of all the sides. For a rectangle: 2 × length + 2 × width.
- Angles in a triangle: 180°. On a straight line: 180°. At a point: 360°. In a quadrilateral: 360°.
- Use equal sides or equal angles (isosceles, square, rhombus) to write an equation without a total.
- After finding x, find what the question asks for, and check it.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Use a fact from geometry to write an equation, solve it by doing the same to both sides, then use the value of x to answer the question.
- Use the perimeter. The perimeter is the total length around the rectangle. A rectangle has two sides of each length.Perimeter = 2(3x + 2) + 2(x + 4) = 36
Equation: 2(3x + 2) + 2(x + 4) = 36 - Expand the brackets. Multiply each term inside the bracket by the number outside it.Expand 2(3x + 2):
2 × 3x = 6x
2 × 2 = 4
Expand 2(x + 4):
2 × x = 2x
2 × 4 = 8
The equation is now 6x + 4 + 2x + 8 = 36 - Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.On the left side: the x terms 6x + 2x make 8x and the numbers 4 + 8 make 12.
The equation is now 8x + 12 = 36 - Subtract 12 from both sides. This undoes the +12, because +12 − 12 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.8x + 12 − 12 = 36 − 12
8x = 24 - Divide both sides by 8. 8x means 8 multiplied by x. The inverse of multiplying by 8 is dividing by 8, and we do it to both sides to keep the balance. This leaves x on its own.8x8 = 248
x = 248 = 3 - Write the solution. x is now on its own, so the equation tells us its value.x = 3
- Answer the question that was asked. The value of x is usually only a stepping stone: read the question again to see what is wanted (a length, an angle, a perimeter) and put the units back.x = 3. The sides are 3 × 3 + 2 = 11 cm and 3 + 4 = 7 cm.
Check. Perimeter = 2 × 11 + 2 × 7 = 36 cm ✓
Watch out. Do not stop at the value of x. Put it back into the expressions on the diagram to find the angle or length that the question asks for.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Use a fact from geometry to write an equation, solve it by doing the same to both sides, then use the value of x to answer the question.
- Use the angles in a triangle. The three angles of a triangle add up to 180°.(x + 20) + 2x + (3x − 20) = 180
Equation: (x + 20) + 2x + (3x − 20) = 180 - Expand the brackets. Multiply each term inside the bracket by the number outside it.(x + 20) has nothing in front of it except +1, so the brackets can simply be removed: x + 20
(3x − 20) has nothing in front of it except +1, so the brackets can simply be removed: 3x − 20
The equation is now x + 20 + 2x + 3x − 20 = 180 - Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.On the left side: the x terms x + 2x + 3x make 6x and the numbers 20 − 20 make 0.
The equation is now 6x = 180 - Divide both sides by 6. 6x means 6 multiplied by x. The inverse of multiplying by 6 is dividing by 6, and we do it to both sides to keep the balance. This leaves x on its own.6x6 = 1806
x = 1806 = 30 - Write the solution. x is now on its own, so the equation tells us its value.x = 30
- Answer the question that was asked. The value of x is usually only a stepping stone: read the question again to see what is wanted (a length, an angle, a perimeter) and put the units back.x = 30. The angles are 50°, 60° and 70°.
Check. 50 + 60 + 70 = 180 ✓
Watch out. Do not stop at the value of x. Put it back into the expressions on the diagram to find the angle or length that the question asks for.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Use a fact from geometry to write an equation, solve it by doing the same to both sides, then use the value of x to answer the question.
- Use angles on a straight line. The angles on a straight line add up to 180°.(2x + 10) + (x + 35) = 180
Equation: (2x + 10) + (x + 35) = 180 - Expand the brackets. Multiply each term inside the bracket by the number outside it.(2x + 10) has nothing in front of it except +1, so the brackets can simply be removed: 2x + 10
(x + 35) has nothing in front of it except +1, so the brackets can simply be removed: x + 35
The equation is now 2x + 10 + x + 35 = 180 - Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.On the left side: the x terms 2x + x make 3x and the numbers 10 + 35 make 45.
The equation is now 3x + 45 = 180 - Subtract 45 from both sides. This undoes the +45, because +45 − 45 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.3x + 45 − 45 = 180 − 45
3x = 135 - Divide both sides by 3. 3x means 3 multiplied by x. The inverse of multiplying by 3 is dividing by 3, and we do it to both sides to keep the balance. This leaves x on its own.3x3 = 1353
x = 1353 = 45 - Write the solution. x is now on its own, so the equation tells us its value.x = 45
- Answer the question that was asked. The value of x is usually only a stepping stone: read the question again to see what is wanted (a length, an angle, a perimeter) and put the units back.x = 45. The angles are 100° and 80°.
Check. 100 + 80 = 180 ✓
Watch out. Do not stop at the value of x. Put it back into the expressions on the diagram to find the angle or length that the question asks for.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Use a fact from geometry to write an equation, solve it by doing the same to both sides, then use the value of x to answer the question.
- Use the equal sides. In an isosceles triangle two sides are equal. The ticks show that AB = AC.AB = AC, so 3x + 1 = x + 9
Equation: 3x + 1 = x + 9 - Remove the x term from the right side: subtract x from both sides. We want all the x terms on one side. The right side has x; subtracting x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.3x + 1 − x = x + 9 − x
2x + 1 = 9 - Subtract 1 from both sides. This undoes the +1, because +1 − 1 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.2x + 1 − 1 = 9 − 1
2x = 8 - Divide both sides by 2. 2x means 2 multiplied by x. The inverse of multiplying by 2 is dividing by 2, and we do it to both sides to keep the balance. This leaves x on its own.2x2 = 82
x = 82 = 4 - Write the solution. x is now on its own, so the equation tells us its value.x = 4
- Answer the question that was asked. The value of x is usually only a stepping stone: read the question again to see what is wanted (a length, an angle, a perimeter) and put the units back.x = 4. AB = AC = 13 cm and BC = 8 cm, so the perimeter is 13 + 13 + 8 = 34 cm.
Check. AB = 3 × 4 + 1 = 13 and AC = 4 + 9 = 13. They are equal ✓
Watch out. Do not stop at the value of x. Put it back into the expressions on the diagram to find the angle or length that the question asks for.
Watch out
Finding x is often not the end. If the question asks for an angle, a length or a perimeter, substitute x back in. And check: three angles of a triangle must add to 180°, so if yours do not, you have made a mistake.Exercise 9H Equations from perimeters and angles
Non-calculatorKey words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Use a fact from geometry to write an equation, solve it by doing the same to both sides, then use the value of x to answer the question.
- Use the perimeter. The perimeter of a rectangle is two lengths plus two widths.Perimeter = 2(2x + 1) + 2(x + 3) = 32
Equation: 2(2x + 1) + 2(x + 3) = 32 - Expand the brackets. Multiply each term inside the bracket by the number outside it.Expand 2(2x + 1):
2 × 2x = 4x
2 × 1 = 2
Expand 2(x + 3):
2 × x = 2x
2 × 3 = 6
The equation is now 4x + 2 + 2x + 6 = 32 - Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.On the left side: the x terms 4x + 2x make 6x and the numbers 2 + 6 make 8.
The equation is now 6x + 8 = 32 - Subtract 8 from both sides. This undoes the +8, because +8 − 8 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.6x + 8 − 8 = 32 − 8
6x = 24 - Divide both sides by 6. 6x means 6 multiplied by x. The inverse of multiplying by 6 is dividing by 6, and we do it to both sides to keep the balance. This leaves x on its own.6x6 = 246
x = 246 = 4 - Write the solution. x is now on its own, so the equation tells us its value.x = 4
- Answer the question that was asked. The value of x is usually only a stepping stone: read the question again to see what is wanted (a length, an angle, a perimeter) and put the units back.x = 4. The sides are 9 cm and 7 cm.
Check. 2 × 9 + 2 × 7 = 32 ✓
Watch out. Do not stop at the value of x. Put it back into the expressions on the diagram to find the angle or length that the question asks for.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Use a fact from geometry to write an equation, solve it by doing the same to both sides, then use the value of x to answer the question.
- Use the perimeter. A square has four equal sides, so the perimeter is 4 times one side.Perimeter = 4(4x − 3) = 52
Equation: 4(4x − 3) = 52 - Expand the brackets. Multiply each term inside the bracket by the number outside it.Expand 4(4x − 3):
4 × 4x = 16x
4 × (−3) = −12
The equation is now 16x − 12 = 52 - Add 12 to both sides. This undoes the −12, because −12 + 12 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.16x − 12 + 12 = 52 + 12
16x = 64 - Divide both sides by 16. 16x means 16 multiplied by x. The inverse of multiplying by 16 is dividing by 16, and we do it to both sides to keep the balance. This leaves x on its own.16x16 = 6416
x = 6416 = 4 - Write the solution. x is now on its own, so the equation tells us its value.x = 4
- Answer the question that was asked. The value of x is usually only a stepping stone: read the question again to see what is wanted (a length, an angle, a perimeter) and put the units back.x = 4. The side of the square is 13 cm.
Check. 4 × 13 = 52 ✓
Watch out. Do not stop at the value of x. Put it back into the expressions on the diagram to find the angle or length that the question asks for.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Use a fact from geometry to write an equation, solve it by doing the same to both sides, then use the value of x to answer the question.
- Use the perimeter. Add up all three sides.Perimeter = (x + 5) + 2x + (2x + 1) = 31
Equation: (x + 5) + 2x + (2x + 1) = 31 - Expand the brackets. Multiply each term inside the bracket by the number outside it.(x + 5) has nothing in front of it except +1, so the brackets can simply be removed: x + 5
(2x + 1) has nothing in front of it except +1, so the brackets can simply be removed: 2x + 1
The equation is now x + 5 + 2x + 2x + 1 = 31 - Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.On the left side: the x terms x + 2x + 2x make 5x and the numbers 5 + 1 make 6.
The equation is now 5x + 6 = 31 - Subtract 6 from both sides. This undoes the +6, because +6 − 6 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.5x + 6 − 6 = 31 − 6
5x = 25 - Divide both sides by 5. 5x means 5 multiplied by x. The inverse of multiplying by 5 is dividing by 5, and we do it to both sides to keep the balance. This leaves x on its own.5x5 = 255
x = 255 = 5 - Write the solution. x is now on its own, so the equation tells us its value.x = 5
- Answer the question that was asked. The value of x is usually only a stepping stone: read the question again to see what is wanted (a length, an angle, a perimeter) and put the units back.x = 5. The sides are 10 cm, 10 cm and 11 cm.
Check. 10 + 10 + 11 = 31 ✓
Watch out. Do not stop at the value of x. Put it back into the expressions on the diagram to find the angle or length that the question asks for.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Use a fact from geometry to write an equation, solve it by doing the same to both sides, then use the value of x to answer the question.
- Use the angles in a triangle. The angles of a triangle add up to 180°.(x + 10) + 2x + (3x + 20) = 180
Equation: (x + 10) + 2x + (3x + 20) = 180 - Expand the brackets. Multiply each term inside the bracket by the number outside it.(x + 10) has nothing in front of it except +1, so the brackets can simply be removed: x + 10
(3x + 20) has nothing in front of it except +1, so the brackets can simply be removed: 3x + 20
The equation is now x + 10 + 2x + 3x + 20 = 180 - Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.On the left side: the x terms x + 2x + 3x make 6x and the numbers 10 + 20 make 30.
The equation is now 6x + 30 = 180 - Subtract 30 from both sides. This undoes the +30, because +30 − 30 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.6x + 30 − 30 = 180 − 30
6x = 150 - Divide both sides by 6. 6x means 6 multiplied by x. The inverse of multiplying by 6 is dividing by 6, and we do it to both sides to keep the balance. This leaves x on its own.6x6 = 1506
x = 1506 = 25 - Write the solution. x is now on its own, so the equation tells us its value.x = 25
- Answer the question that was asked. The value of x is usually only a stepping stone: read the question again to see what is wanted (a length, an angle, a perimeter) and put the units back.x = 25. The angles are 35°, 50° and 95°. The largest is 95°.
Check. 35 + 50 + 95 = 180 ✓
Watch out. Do not stop at the value of x. Put it back into the expressions on the diagram to find the angle or length that the question asks for.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Use a fact from geometry to write an equation, solve it by doing the same to both sides, then use the value of x to answer the question.
- Use angles on a straight line. Angles on a straight line add up to 180°.(3x + 10) + (2x + 20) = 180
Equation: (3x + 10) + (2x + 20) = 180 - Expand the brackets. Multiply each term inside the bracket by the number outside it.(3x + 10) has nothing in front of it except +1, so the brackets can simply be removed: 3x + 10
(2x + 20) has nothing in front of it except +1, so the brackets can simply be removed: 2x + 20
The equation is now 3x + 10 + 2x + 20 = 180 - Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.On the left side: the x terms 3x + 2x make 5x and the numbers 10 + 20 make 30.
The equation is now 5x + 30 = 180 - Subtract 30 from both sides. This undoes the +30, because +30 − 30 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.5x + 30 − 30 = 180 − 30
5x = 150 - Divide both sides by 5. 5x means 5 multiplied by x. The inverse of multiplying by 5 is dividing by 5, and we do it to both sides to keep the balance. This leaves x on its own.5x5 = 1505
x = 1505 = 30 - Write the solution. x is now on its own, so the equation tells us its value.x = 30
- Answer the question that was asked. The value of x is usually only a stepping stone: read the question again to see what is wanted (a length, an angle, a perimeter) and put the units back.x = 30. The angles are 100° and 80°.
Check. 100 + 80 = 180 ✓
Watch out. Do not stop at the value of x. Put it back into the expressions on the diagram to find the angle or length that the question asks for.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Use a fact from geometry to write an equation, solve it by doing the same to both sides, then use the value of x to answer the question.
- Use angles round a point. Angles around a point add up to 360°.2x + (3x + 10) + (x + 50) = 360
Equation: 2x + (3x + 10) + (x + 50) = 360 - Expand the brackets. Multiply each term inside the bracket by the number outside it.(3x + 10) has nothing in front of it except +1, so the brackets can simply be removed: 3x + 10
(x + 50) has nothing in front of it except +1, so the brackets can simply be removed: x + 50
The equation is now 2x + 3x + 10 + x + 50 = 360 - Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.On the left side: the x terms 2x + 3x + x make 6x and the numbers 10 + 50 make 60.
The equation is now 6x + 60 = 360 - Subtract 60 from both sides. This undoes the +60, because +60 − 60 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.6x + 60 − 60 = 360 − 60
6x = 300 - Divide both sides by 6. 6x means 6 multiplied by x. The inverse of multiplying by 6 is dividing by 6, and we do it to both sides to keep the balance. This leaves x on its own.6x6 = 3006
x = 3006 = 50 - Write the solution. x is now on its own, so the equation tells us its value.x = 50
- Answer the question that was asked. The value of x is usually only a stepping stone: read the question again to see what is wanted (a length, an angle, a perimeter) and put the units back.x = 50. The angles are 100°, 160° and 100°.
Check. 100 + 160 + 100 = 360 ✓
Watch out. Do not stop at the value of x. Put it back into the expressions on the diagram to find the angle or length that the question asks for.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Use a fact from geometry to write an equation, solve it by doing the same to both sides, then use the value of x to answer the question.
- Use the angles in a quadrilateral. The angles of a quadrilateral add up to 360°.x + (x + 30) + (2x + 10) + (2x + 20) = 360
Equation: x + (x + 30) + (2x + 10) + (2x + 20) = 360 - Expand the brackets. Multiply each term inside the bracket by the number outside it.(x + 30) has nothing in front of it except +1, so the brackets can simply be removed: x + 30
(2x + 10) has nothing in front of it except +1, so the brackets can simply be removed: 2x + 10
(2x + 20) has nothing in front of it except +1, so the brackets can simply be removed: 2x + 20
The equation is now x + x + 30 + 2x + 10 + 2x + 20 = 360 - Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.On the left side: the x terms x + x + 2x + 2x make 6x and the numbers 30 + 10 + 20 make 60.
The equation is now 6x + 60 = 360 - Subtract 60 from both sides. This undoes the +60, because +60 − 60 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.6x + 60 − 60 = 360 − 60
6x = 300 - Divide both sides by 6. 6x means 6 multiplied by x. The inverse of multiplying by 6 is dividing by 6, and we do it to both sides to keep the balance. This leaves x on its own.6x6 = 3006
x = 3006 = 50 - Write the solution. x is now on its own, so the equation tells us its value.x = 50
- Answer the question that was asked. The value of x is usually only a stepping stone: read the question again to see what is wanted (a length, an angle, a perimeter) and put the units back.x = 50. The angles are 50°, 80°, 110° and 120°.
Check. 50 + 80 + 110 + 120 = 360 ✓
Watch out. Do not stop at the value of x. Put it back into the expressions on the diagram to find the angle or length that the question asks for.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Use a fact from geometry to write an equation, solve it by doing the same to both sides, then use the value of x to answer the question.
- Use the base angles and the angle sum. In an isosceles triangle the two base angles are equal. The angles of a triangle add up to 180°.(2x + 5) + (2x + 5) + 50 = 180
Equation: (2x + 5) + (2x + 5) + 50 = 180 - Expand the brackets. Multiply each term inside the bracket by the number outside it.(2x + 5) has nothing in front of it except +1, so the brackets can simply be removed: 2x + 5
(2x + 5) has nothing in front of it except +1, so the brackets can simply be removed: 2x + 5
The equation is now 2x + 5 + 2x + 5 + 50 = 180 - Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.On the left side: the x terms 2x + 2x make 4x and the numbers 5 + 5 + 50 make 60.
The equation is now 4x + 60 = 180 - Subtract 60 from both sides. This undoes the +60, because +60 − 60 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.4x + 60 − 60 = 180 − 60
4x = 120 - Divide both sides by 4. 4x means 4 multiplied by x. The inverse of multiplying by 4 is dividing by 4, and we do it to both sides to keep the balance. This leaves x on its own.4x4 = 1204
x = 1204 = 30 - Write the solution. x is now on its own, so the equation tells us its value.x = 30
- Answer the question that was asked. The value of x is usually only a stepping stone: read the question again to see what is wanted (a length, an angle, a perimeter) and put the units back.x = 30. Each base angle is 65°.
Check. 65 + 65 + 50 = 180 ✓
Watch out. Do not stop at the value of x. Put it back into the expressions on the diagram to find the angle or length that the question asks for.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Use a fact from geometry to write an equation, solve it by doing the same to both sides, then use the value of x to answer the question.
- Use the equal sides. The ticks show that AB = AC.2x + 5 = 4x − 3
Equation: 2x + 5 = 4x − 3 - Remove the x term from the left side: subtract 2x from both sides. We want all the x terms on one side. The left side has 2x; subtracting 2x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.2x + 5 − 2x = 4x − 3 − 2x
5 = 2x − 3 - Add 3 to both sides. This undoes the −3, because −3 + 3 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.5 + 3 = 2x − 3 + 3
8 = 2x - Divide both sides by 2. 2x means 2 multiplied by x. The inverse of multiplying by 2 is dividing by 2, and we do it to both sides to keep the balance. This leaves x on its own.82 = 2x2
x = 82 = 4 - Write the solution. x is now on its own, so the equation tells us its value.x = 4
- Answer the question that was asked. The value of x is usually only a stepping stone: read the question again to see what is wanted (a length, an angle, a perimeter) and put the units back.x = 4. AB = AC = 13 cm, and BC = 10 cm, so the perimeter is 36 cm.
Check. 2 × 4 + 5 = 13 and 4 × 4 − 3 = 13 ✓
Watch out. Do not stop at the value of x. Put it back into the expressions on the diagram to find the angle or length that the question asks for.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Use a fact from geometry to write an equation, solve it by doing the same to both sides, then use the value of x to answer the question.
- Use the equal sides. All four sides of a rhombus are equal, so the two expressions are equal.3x − 1 = x + 9
Equation: 3x − 1 = x + 9 - Remove the x term from the right side: subtract x from both sides. We want all the x terms on one side. The right side has x; subtracting x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.3x − 1 − x = x + 9 − x
2x − 1 = 9 - Add 1 to both sides. This undoes the −1, because −1 + 1 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.2x − 1 + 1 = 9 + 1
2x = 10 - Divide both sides by 2. 2x means 2 multiplied by x. The inverse of multiplying by 2 is dividing by 2, and we do it to both sides to keep the balance. This leaves x on its own.2x2 = 102
x = 102 = 5 - Write the solution. x is now on its own, so the equation tells us its value.x = 5
- Answer the question that was asked. The value of x is usually only a stepping stone: read the question again to see what is wanted (a length, an angle, a perimeter) and put the units back.x = 5. Each side is 14 cm, so the perimeter is 4 × 14 = 56 cm.
Check. 3 × 5 − 1 = 14 and 5 + 9 = 14 ✓
Watch out. Do not stop at the value of x. Put it back into the expressions on the diagram to find the angle or length that the question asks for.
Method. Use the angle sum to find x, then look at the three angles.
- Write an equation. The angles of a triangle add up to 180°.x + 2x + 3x = 180
- Collect like terms. 1x + 2x + 3x = 6x.6x = 180
- Divide both sides by 6. x = 180 ÷ 6 = 30
- Find the three angles. Put x = 30 into each expression.30°, 2 × 30 = 60°, 3 × 30 = 90°
- Decide. A right angle is 90°.One angle is 90°, so Zane is correct.
Check. 30 + 60 + 90 = 180. ✓
Watch out. In a “show that” or “is he correct?” question, finish with a sentence that gives the conclusion.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Use a fact from geometry to write an equation, solve it by doing the same to both sides, then use the value of x to answer the question.
- Use the angles in a triangle. The angles of a triangle add up to 180°.(2x − 10) + (x + 30) + (3x − 20) = 180
Equation: (2x − 10) + (x + 30) + (3x − 20) = 180 - Expand the brackets. Multiply each term inside the bracket by the number outside it.(2x − 10) has nothing in front of it except +1, so the brackets can simply be removed: 2x − 10
(x + 30) has nothing in front of it except +1, so the brackets can simply be removed: x + 30
(3x − 20) has nothing in front of it except +1, so the brackets can simply be removed: 3x − 20
The equation is now 2x − 10 + x + 30 + 3x − 20 = 180 - Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.On the left side: the x terms 2x + x + 3x make 6x and the numbers −10 + 30 − 20 make 0.
The equation is now 6x = 180 - Divide both sides by 6. 6x means 6 multiplied by x. The inverse of multiplying by 6 is dividing by 6, and we do it to both sides to keep the balance. This leaves x on its own.6x6 = 1806
x = 1806 = 30 - Write the solution. x is now on its own, so the equation tells us its value.x = 30
- Answer the question that was asked. The value of x is usually only a stepping stone: read the question again to see what is wanted (a length, an angle, a perimeter) and put the units back.x = 30. The angles are 50°, 60° and 70°, so the largest is 70°.
Check. 50 + 60 + 70 = 180 ✓
Watch out. Do not stop at the value of x. Put it back into the expressions on the diagram to find the angle or length that the question asks for.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Use a fact from geometry to write an equation, solve it by doing the same to both sides, then use the value of x to answer the question.
- Use the angles on a straight line. Angles on a straight line add up to 180°.(x + 40) + (2x + 20) = 180
Equation: (x + 40) + (2x + 20) = 180 - Expand the brackets. Multiply each term inside the bracket by the number outside it.(x + 40) has nothing in front of it except +1, so the brackets can simply be removed: x + 40
(2x + 20) has nothing in front of it except +1, so the brackets can simply be removed: 2x + 20
The equation is now x + 40 + 2x + 20 = 180 - Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.On the left side: the x terms x + 2x make 3x and the numbers 40 + 20 make 60.
The equation is now 3x + 60 = 180 - Subtract 60 from both sides. This undoes the +60, because +60 − 60 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.3x + 60 − 60 = 180 − 60
3x = 120 - Divide both sides by 3. 3x means 3 multiplied by x. The inverse of multiplying by 3 is dividing by 3, and we do it to both sides to keep the balance. This leaves x on its own.3x3 = 1203
x = 1203 = 40 - Write the solution. x is now on its own, so the equation tells us its value.x = 40
- Answer the question that was asked. The value of x is usually only a stepping stone: read the question again to see what is wanted (a length, an angle, a perimeter) and put the units back.x = 40. The angles are 80° and 100°.
Check. 80 + 100 = 180 ✓
Watch out. Do not stop at the value of x. Put it back into the expressions on the diagram to find the angle or length that the question asks for.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Use a fact from geometry to write an equation, solve it by doing the same to both sides, then use the value of x to answer the question.
- Use the perimeter. A rectangle has two lengths and two widths.2x + 2 × 4 = 20
Equation: 2x + 8 = 20 - Subtract 8 from both sides. This undoes the +8, because +8 − 8 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.2x + 8 − 8 = 20 − 8
2x = 12 - Divide both sides by 2. 2x means 2 multiplied by x. The inverse of multiplying by 2 is dividing by 2, and we do it to both sides to keep the balance. This leaves x on its own.2x2 = 122
x = 122 = 6 - Write the solution. x is now on its own, so the equation tells us its value.x = 6
- Answer the question that was asked. The value of x is usually only a stepping stone: read the question again to see what is wanted (a length, an angle, a perimeter) and put the units back.x = 6, so the length is 6 cm.
Check. 2 × 6 + 2 × 4 = 20 ✓
Watch out. Do not stop at the value of x. Put it back into the expressions on the diagram to find the angle or length that the question asks for.