GCSE Maths: Constructing triangles (Higher)
Part of the GCSE Maths loci, bearings and vectors chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Constructing triangles
A triangle has three sides and three angles, but you do not need all six to draw it. If you are told enough information, there is only one possible triangle (apart from its position on the page). In Chapter 21 you met the same ideas as the conditions for congruent triangles: three sides (SSS), two sides and the angle between them (SAS), and two angles and a side (ASA).
SSS: three sides. Draw one side as the base, say AB. The corner C must be the right distance from A and from B. Draw an arc of the right radius from A, and another from B. The arcs cross at C.
SAS: two sides and the angle between them. Draw one side, measure the angle at one end with a protractor, mark the second side along the new line, then join the ends.
ASA: two angles and a side. Draw the side, then measure one angle at each end. The two lines cross at the third corner.
Right-angled triangles (RHS). If you know the hypotenuse and one other side, construct the right angle, then use an arc for the hypotenuse.
When is a triangle impossible? The two shorter sides must be able to reach each other. If the arcs do not cross, there is no triangle. This happens when the two short sides add up to less than or equal to the longest side. For example 3 cm, 4 cm and 8 cm cannot make a triangle because 3 + 4 = 7 is less than 8: the arcs fall short of each other.
Accuracy. A good construction is accurate to about 1 mm for lengths and 1° for angles. Always say which measurement you have taken and give a sensible unit. Leave all construction arcs on your page.
Key ideas
- SSS: base, then an arc from each end. SAS: side, angle at one end, then second side. ASA: side, then an angle at each end.
- Three angles alone do not fix a triangle: you can have a small or a big one with the same angles.
- A triangle with sides a, b, c is only possible if the two shorter sides add up to more than the longest.
- Measure carefully and leave construction arcs.
- RHS: build the right angle first, then use an arc for the hypotenuse.
Key words. To construct means to draw accurately using only a ruler and a pair of compasses (no protractor). The construction arcs are the curved lines the compasses leave; you must not rub them out.
Method. Draw the longest side as a base, then find the third corner with two arcs. Measure the angle with a protractor.
- Draw the base AB. Choose one side as the base and draw it with a ruler.AB = 7 cm
- Draw an arc from A. C is 6 cm from A, so every possible position of C is on an arc of radius 6 cm centred on A.Compasses set to 6 cm, point on A.
- Draw an arc from B. C is also 5 cm from B. Where the two arcs cross is the only point that is the right distance from both A and B.Compasses set to 5 cm, point on B. The arcs cross at C.
- Join AC and BC. Use a ruler. Keep the arcs.
- Measure angle C. The biggest angle is opposite the longest side (7 cm), which is angle C. Use a protractor.Angle C = 78.5° to 1 decimal place, so about 78°
Check. Cosine rule: cos C = (52 + 62 − 72) ÷ (2 × 5 × 6) = 0.2000, so C = 78.5°. Your measurement should be within 1° of this. ✓
Watch out. The two arcs must have the radii of the two sides that meet at C. Do not swap them.
Key words. To construct means to draw accurately using only a ruler and a pair of compasses (no protractor). The construction arcs are the curved lines the compasses leave; you must not rub them out.
Method. Draw one side, measure the angle at one end of it, mark the second side along the new line, then join up.
- Draw the side AB. It is one of the two sides next to the known angle.AB = 6 cm
- Measure the angle at A. Place the centre of the protractor on A with its zero line along AB. Mark the angle and draw a faint line through the mark.Angle A = 50°
- Mark C. C is on the new line and AC is a given length. Measure with a ruler (or an arc of the compasses).AC = 4.5 cm
- Join BC and measure it. This completes the triangle. Measure BC with a ruler.BC = 4.64 cm ≈ 4.6 cm
Check. Cosine rule: BC2 = 62 + 4.52 − 2 × 6 × 4.5 × cos 50° = 21.54, so BC = 4.64 cm. Your measurement should be within 1 mm of this. ✓
Watch out. The angle must be between the two given sides. If it is not between them, you cannot draw the triangle this way.
Key words. To construct means to draw accurately using only a ruler and a pair of compasses (no protractor). The construction arcs are the curved lines the compasses leave; you must not rub them out.
Method. Draw the side, then draw the angle at each end. The lines meet at the third corner.
- Draw the side AB. The side between the two known angles is the base.AB = 6 cm
- Draw the angle at A. Use a protractor at A with its zero line along AB. Draw a long faint line.Angle A = 50°
- Draw the angle at B. Use the protractor at B. Measure from BA (the zero line points back towards A). Draw a line that crosses the first line at C.Angle B = 60°
- Measure the side you want. Where the two lines cross is the third corner C. Measure with a ruler.AC = 5.53 cm ≈ 5.5 cm
Check. Sine rule: the third angle is 180° − 50° − 60° = 70°, and AC = 6 × sin 60° ÷ sin 70° = 5.53 cm. ✓
Watch out. Angles in a triangle add up to 180°, so the angles must add up to less than 180° or the lines will never meet.
Key words. To construct means to draw accurately using only a ruler and a pair of compasses (no protractor). The construction arcs are the curved lines the compasses leave; you must not rub them out.
Method. Try to draw it. The two shorter sides must reach each other when they are attached to the ends of the longest side.
- Draw the longest side. Take the 8 cm side as the base AB.AB = 8 cm
- Draw the two arcs. C must be 3 cm from A and 4 cm from B. So the arcs have radii 3 cm and 4 cm.Arc from A: radius 3 cm. Arc from B: radius 4 cm.
- See what happens. The arcs reach 3 cm and 4 cm from the ends. Together they cover only 3 + 4 = 7 cm of the 8 cm base.3 + 4 = 7 and 7 < 8, so the arcs do not meet.
- Write the conclusion. If the arcs never cross there is no point C.No triangle can be drawn. The two shorter sides must add up to more than the longest side.
Check. A triangle with sides 3, 4, 5 does exist, because 3 + 4 = 7 is more than 5. ✓
Watch out. It is not enough for the sides to be different. They must satisfy: two shorter sides add up to more than the longest.
Key words. To construct means to draw accurately using only a ruler and a pair of compasses (no protractor). The construction arcs are the curved lines the compasses leave; you must not rub them out.
Method. A right angle, the hypotenuse and one other side fix a triangle. Construct the right angle first, then the arc for the hypotenuse.
- Draw the base AB. The base is one of the shorter sides.AB = 2.5 cm
- Construct a right angle at B. Use the perpendicular-at-a-point construction at B (arcs from B, then larger arcs). The perpendicular line goes up from B.Angle ABC = 90°
- Draw an arc from A. C is 6.5 cm from A, so set the compasses to 6.5 cm and draw an arc from A that crosses the perpendicular at C.AC = 6.5 cm
- Join AC and measure BC. Use a ruler.BC = 6.0 cm
Check. Pythagoras: BC2 = 6.52 − 2.52 = 36.00, so BC = 6.0 cm. ✓
Watch out. The hypotenuse is opposite the right angle, so the arc is drawn from the end of the base that is not at the right angle.
Watch out
Check which sides and angles you have been given. For SAS the angle must be between the two sides. If you use the wrong end of the line you will draw a different triangle. And three angles (AAA) is not enough: it only tells you the shape, not the size.Exercise 22D Constructing triangles and measuring
Calculator allowedKey words. To construct means to draw accurately using only a ruler and a pair of compasses (no protractor). The construction arcs are the curved lines the compasses leave; you must not rub them out.
Method. Check for the condition for a unique triangle.
- Name the information. Three sides is SSS.SSS
- Check they fit. The two shorter sides must add to more than the longest.5 + 6 = 11 > 7
- Write the answer. Yes. SSS fixes the triangle.
Check. The arcs of radius 6 and 5 do cross. ✓
Key words. To construct means to draw accurately using only a ruler and a pair of compasses (no protractor). The construction arcs are the curved lines the compasses leave; you must not rub them out.
Method. Angles give the shape but not the size.
- Add the angles. They must add to 180°.40 + 60 + 80 = 180°. They do make a triangle shape.
- Think about the size. You can draw a small triangle or a larger one with the same three angles.Any enlargement keeps the angles.
- Write the answer. No. Many different-sized triangles have these angles.
Check. A 1 cm and a 2 cm version both have 40°, 60°, 80°. ✓
Key words. To construct means to draw accurately using only a ruler and a pair of compasses (no protractor). The construction arcs are the curved lines the compasses leave; you must not rub them out.
Method. Is the angle between the two sides?
- Check the sides. Angle A is at the vertex where AB and AC meet.The angle is between the two given sides.
- Write the answer. This is SAS.Yes. SAS fixes the triangle.
Check. Draw AB, the angle at A, and mark C 4 cm along. ✓
Key words. To construct means to draw accurately using only a ruler and a pair of compasses (no protractor). The construction arcs are the curved lines the compasses leave; you must not rub them out.
Method. Check that the two shorter sides can reach each other.
- Add the shorter sides. They must add up to more than the longest side.3 + 4 = 7
- Compare. 7 < 8, so the arcs will not meet
- Write the answer. No, no such triangle exists.
Check. 3, 4, 5 does work since 3 + 4 > 5. ✓
Key words. To construct means to draw accurately using only a ruler and a pair of compasses (no protractor). The construction arcs are the curved lines the compasses leave; you must not rub them out.
Method. Draw the longest side as a base, then find the third corner with two arcs. Measure the angle with a protractor.
- Draw the base AB. Choose one side as the base and draw it with a ruler.AB = 6 cm
- Draw an arc from A. C is 5 cm from A, so every possible position of C is on an arc of radius 5 cm centred on A.Compasses set to 5 cm, point on A.
- Draw an arc from B. C is also 4 cm from B. Where the two arcs cross is the only point that is the right distance from both A and B.Compasses set to 4 cm, point on B. The arcs cross at C.
- Join AC and BC. Use a ruler. Keep the arcs.
- Measure angle C. The biggest angle is opposite the longest side (6 cm), which is angle C. Use a protractor.Angle C = 82.8° to 1 decimal place, so about 83°
Check. Cosine rule: cos C = (42 + 52 − 62) ÷ (2 × 4 × 5) = 0.1250, so C = 82.8°. Your measurement should be within 1° of this. ✓
Watch out. The two arcs must have the radii of the two sides that meet at C. Do not swap them.
Key words. To construct means to draw accurately using only a ruler and a pair of compasses (no protractor). The construction arcs are the curved lines the compasses leave; you must not rub them out.
Method. Draw the longest side as a base, then find the third corner with two arcs. Measure the angle with a protractor.
- Draw the base AB. Choose one side as the base and draw it with a ruler.AB = 6 cm
- Draw an arc from A. C is 6 cm from A, so every possible position of C is on an arc of radius 6 cm centred on A.Compasses set to 6 cm, point on A.
- Draw an arc from B. C is also 6 cm from B. Where the two arcs cross is the only point that is the right distance from both A and B.Compasses set to 6 cm, point on B. The arcs cross at C.
- Join AC and BC. Use a ruler. Keep the arcs.
- Measure angle A. The biggest angle is opposite the longest side (6 cm), which is angle A. Use a protractor.Angle A = 60.0° to 1 decimal place, so about 60°
Check. Cosine rule: cos A = (62 + 62 − 62) ÷ (2 × 6 × 6) = 0.5000, so A = 60.0°. Your measurement should be within 1° of this. ✓
Watch out. The two arcs must have the radii of the two sides that meet at C. Do not swap them.
Key words. To construct means to draw accurately using only a ruler and a pair of compasses (no protractor). The construction arcs are the curved lines the compasses leave; you must not rub them out.
Method. Draw the longest side as a base, then find the third corner with two arcs. Measure the angle with a protractor.
- Draw the base AB. Choose one side as the base and draw it with a ruler.AB = 5 cm
- Draw an arc from A. C is 4 cm from A, so every possible position of C is on an arc of radius 4 cm centred on A.Compasses set to 4 cm, point on A.
- Draw an arc from B. C is also 3 cm from B. Where the two arcs cross is the only point that is the right distance from both A and B.Compasses set to 3 cm, point on B. The arcs cross at C.
- Join AC and BC. Use a ruler. Keep the arcs.
- Measure angle C. The biggest angle is opposite the longest side (5 cm), which is angle C. Use a protractor.Angle C = 90.0° to 1 decimal place, so about 90°
Check. Cosine rule: cos C = (32 + 42 − 52) ÷ (2 × 3 × 4) = 0.0000, so C = 90.0°. Your measurement should be within 1° of this. ✓
Watch out. The two arcs must have the radii of the two sides that meet at C. Do not swap them.
Key words. To construct means to draw accurately using only a ruler and a pair of compasses (no protractor). The construction arcs are the curved lines the compasses leave; you must not rub them out.
Method. Draw the longest side as a base, then find the third corner with two arcs. Measure the angle with a protractor.
- Draw the base AB. Choose one side as the base and draw it with a ruler.AB = 8 cm
- Draw an arc from A. C is 5 cm from A, so every possible position of C is on an arc of radius 5 cm centred on A.Compasses set to 5 cm, point on A.
- Draw an arc from B. C is also 5 cm from B. Where the two arcs cross is the only point that is the right distance from both A and B.Compasses set to 5 cm, point on B. The arcs cross at C.
- Join AC and BC. Use a ruler. Keep the arcs.
- Measure angle C. The biggest angle is opposite the longest side (8 cm), which is angle C. Use a protractor.Angle C = 106.3° to 1 decimal place, so about 106°
Check. Cosine rule: cos C = (52 + 52 − 82) ÷ (2 × 5 × 5) = −0.2800, so C = 106.3°. Your measurement should be within 1° of this. ✓
Watch out. The two arcs must have the radii of the two sides that meet at C. Do not swap them.
Key words. To construct means to draw accurately using only a ruler and a pair of compasses (no protractor). The construction arcs are the curved lines the compasses leave; you must not rub them out.
Method. Draw one side, measure the angle at one end of it, mark the second side along the new line, then join up.
- Draw the side AB. It is one of the two sides next to the known angle.AB = 7 cm
- Measure the angle at A. Place the centre of the protractor on A with its zero line along AB. Mark the angle and draw a faint line through the mark.Angle A = 40°
- Mark C. C is on the new line and AC is a given length. Measure with a ruler (or an arc of the compasses).AC = 5 cm
- Join BC and measure it. This completes the triangle. Measure BC with a ruler.BC = 4.51 cm ≈ 4.5 cm
Check. Cosine rule: BC2 = 72 + 52 − 2 × 7 × 5 × cos 40° = 20.38, so BC = 4.51 cm. Your measurement should be within 1 mm of this. ✓
Watch out. The angle must be between the two given sides. If it is not between them, you cannot draw the triangle this way.
Key words. To construct means to draw accurately using only a ruler and a pair of compasses (no protractor). The construction arcs are the curved lines the compasses leave; you must not rub them out.
Method. Draw one side, measure the angle at one end of it, mark the second side along the new line, then join up.
- Draw the side AB. It is one of the two sides next to the known angle.AB = 6 cm
- Measure the angle at A. Place the centre of the protractor on A with its zero line along AB. Mark the angle and draw a faint line through the mark.Angle A = 90°
- Mark C. C is on the new line and AC is a given length. Measure with a ruler (or an arc of the compasses).AC = 8 cm
- Join BC and measure it. This completes the triangle. Measure BC with a ruler.BC = 10.00 cm ≈ 10.0 cm
Check. Cosine rule: BC2 = 62 + 82 − 2 × 6 × 8 × cos 90° = 100.00, so BC = 10.00 cm. Your measurement should be within 1 mm of this. ✓
Watch out. The angle must be between the two given sides. If it is not between them, you cannot draw the triangle this way.
Key words. To construct means to draw accurately using only a ruler and a pair of compasses (no protractor). The construction arcs are the curved lines the compasses leave; you must not rub them out.
Method. Draw one side, measure the angle at one end of it, mark the second side along the new line, then join up.
- Draw the side AB. It is one of the two sides next to the known angle.AB = 5 cm
- Measure the angle at A. Place the centre of the protractor on A with its zero line along AB. Mark the angle and draw a faint line through the mark.Angle A = 60°
- Mark C. C is on the new line and AC is a given length. Measure with a ruler (or an arc of the compasses).AC = 5 cm
- Join BC and measure it. This completes the triangle. Measure BC with a ruler.BC = 5.00 cm ≈ 5.0 cm
Check. Cosine rule: BC2 = 52 + 52 − 2 × 5 × 5 × cos 60° = 25.00, so BC = 5.00 cm. Your measurement should be within 1 mm of this. ✓
Watch out. The angle must be between the two given sides. If it is not between them, you cannot draw the triangle this way.
Key words. To construct means to draw accurately using only a ruler and a pair of compasses (no protractor). The construction arcs are the curved lines the compasses leave; you must not rub them out.
Method. Draw one side, measure the angle at one end of it, mark the second side along the new line, then join up.
- Draw the side AB. It is one of the two sides next to the known angle.AB = 8 cm
- Measure the angle at A. Place the centre of the protractor on A with its zero line along AB. Mark the angle and draw a faint line through the mark.Angle A = 120°
- Mark C. C is on the new line and AC is a given length. Measure with a ruler (or an arc of the compasses).AC = 6 cm
- Join BC and measure it. This completes the triangle. Measure BC with a ruler.BC = 12.17 cm ≈ 12.2 cm
Check. Cosine rule: BC2 = 82 + 62 − 2 × 8 × 6 × cos 120° = 148.00, so BC = 12.17 cm. Your measurement should be within 1 mm of this. ✓
Watch out. The angle must be between the two given sides. If it is not between them, you cannot draw the triangle this way.
Key words. To construct means to draw accurately using only a ruler and a pair of compasses (no protractor). The construction arcs are the curved lines the compasses leave; you must not rub them out.
Method. Draw the side, then draw the angle at each end. The lines meet at the third corner.
- Draw the side AB. The side between the two known angles is the base.AB = 7 cm
- Draw the angle at A. Use a protractor at A with its zero line along AB. Draw a long faint line.Angle A = 40°
- Draw the angle at B. Use the protractor at B. Measure from BA (the zero line points back towards A). Draw a line that crosses the first line at C.Angle B = 70°
- Measure the side you want. Where the two lines cross is the third corner C. Measure with a ruler.AC = 7.00 cm ≈ 7.0 cm
Check. Sine rule: the third angle is 180° − 40° − 70° = 70°, and AC = 7 × sin 70° ÷ sin 70° = 7.00 cm. ✓
Watch out. Angles in a triangle add up to 180°, so the angles must add up to less than 180° or the lines will never meet.
Key words. To construct means to draw accurately using only a ruler and a pair of compasses (no protractor). The construction arcs are the curved lines the compasses leave; you must not rub them out.
Method. Draw the side, then draw the angle at each end. The lines meet at the third corner.
- Draw the side AB. The side between the two known angles is the base.AB = 6 cm
- Draw the angle at A. Use a protractor at A with its zero line along AB. Draw a long faint line.Angle A = 60°
- Draw the angle at B. Use the protractor at B. Measure from BA (the zero line points back towards A). Draw a line that crosses the first line at C.Angle B = 60°
- Measure the side you want. Where the two lines cross is the third corner C. Measure with a ruler.AC = 6.00 cm ≈ 6.0 cm
Check. Sine rule: the third angle is 180° − 60° − 60° = 60°, and AC = 6 × sin 60° ÷ sin 60° = 6.00 cm. ✓
Watch out. Angles in a triangle add up to 180°, so the angles must add up to less than 180° or the lines will never meet.
Key words. To construct means to draw accurately using only a ruler and a pair of compasses (no protractor). The construction arcs are the curved lines the compasses leave; you must not rub them out.
Method. Draw the side, then draw the angle at each end. The lines meet at the third corner.
- Draw the side AB. The side between the two known angles is the base.AB = 8 cm
- Draw the angle at A. Use a protractor at A with its zero line along AB. Draw a long faint line.Angle A = 30°
- Draw the angle at B. Use the protractor at B. Measure from BA (the zero line points back towards A). Draw a line that crosses the first line at C.Angle B = 45°
- Measure the side you want. Where the two lines cross is the third corner C. Measure with a ruler.AC = 5.86 cm ≈ 5.9 cm
Check. Sine rule: the third angle is 180° − 30° − 45° = 105°, and AC = 8 × sin 45° ÷ sin 105° = 5.86 cm. ✓
Watch out. Angles in a triangle add up to 180°, so the angles must add up to less than 180° or the lines will never meet.
Key words. To construct means to draw accurately using only a ruler and a pair of compasses (no protractor). The construction arcs are the curved lines the compasses leave; you must not rub them out.
Method. A right angle, the hypotenuse and one other side fix a triangle. Construct the right angle first, then the arc for the hypotenuse.
- Draw the base AB. The base is one of the shorter sides.AB = 6 cm
- Construct a right angle at B. Use the perpendicular-at-a-point construction at B (arcs from B, then larger arcs). The perpendicular line goes up from B.Angle ABC = 90°
- Draw an arc from A. C is 10 cm from A, so set the compasses to 10 cm and draw an arc from A that crosses the perpendicular at C.AC = 10 cm
- Join AC and measure BC. Use a ruler.BC = 8.0 cm
Check. Pythagoras: BC2 = 102 − 62 = 64.00, so BC = 8.0 cm. ✓
Watch out. The hypotenuse is opposite the right angle, so the arc is drawn from the end of the base that is not at the right angle.
Key words. To construct means to draw accurately using only a ruler and a pair of compasses (no protractor). The construction arcs are the curved lines the compasses leave; you must not rub them out.
Method. Use SSS with the base BC = 4 cm and arcs of radius 6 cm from B and from C.
- Draw the base BC. The unequal side makes a good base.BC = 4 cm
- Draw arcs from B and C. Both arcs have radius 6 cm, because AB = AC = 6 cm. They cross at A.Compass setting 6 cm.
- Join AB and AC and measure. Use a protractor at A, then at B.Angle A = 38.9° to 1 d.p.
Base angle = (180° − 38.9°) ÷ 2 = 70.5°
Check. The three angles add up: 39 + 2 × 71 = 180. ✓
Watch out. In an isosceles triangle the two angles beside the unequal side are equal.
Key words. To construct means to draw accurately using only a ruler and a pair of compasses (no protractor). The construction arcs are the curved lines the compasses leave; you must not rub them out.
Method. Change the real lengths to lengths on the drawing, then use SSS.
- Change the lengths to the drawing. 1 cm stands for 10 m, so divide by 10.40 m → 4 cm, 50 m → 5 cm, 60 m → 6 cm
- Construct the triangle. Draw the 6 cm side as a base, then arcs of radius 4 cm and 5 cm from its ends.See the steps in the example.
- Measure the largest angle. It is opposite the longest side (60 m).About 83°
Check. Cosine rule: cos(angle) = (402 + 502 − 602) ÷ (2 × 40 × 50) = 0.125, angle = 82.8°. ✓
Watch out. The scale means every length is divided by 10 for the drawing. Angles do not change.
Key words. To construct means to draw accurately using only a ruler and a pair of compasses (no protractor). The construction arcs are the curved lines the compasses leave; you must not rub them out.
Method. Draw the side, then draw the angle at each end. The lines meet at the third corner.
- Draw the side AB. The side between the two known angles is the base.AB = 6 cm
- Draw the angle at A. Use a protractor at A with its zero line along AB. Draw a long faint line.Angle A = 50°
- Draw the angle at B. Use the protractor at B. Measure from BA (the zero line points back towards A). Draw a line that crosses the first line at C.Angle B = 60°
- Measure the side you want. Where the two lines cross is the third corner C. Measure with a ruler.AC = 5.53 cm ≈ 5.5 cm
Check. Sine rule: the third angle is 180° − 50° − 60° = 70°, and AC = 6 × sin 60° ÷ sin 70° = 5.53 cm. ✓
Watch out. Angles in a triangle add up to 180°, so the angles must add up to less than 180° or the lines will never meet.
Key words. To construct means to draw accurately using only a ruler and a pair of compasses (no protractor). The construction arcs are the curved lines the compasses leave; you must not rub them out.
Method. Check whether the two shorter sides can reach each other.
- Add the two shorter sides. In a real triangle the two shorter sides together must be longer than the longest side.4 + 5 = 9
- Compare with the longest side. 9 < 10
- Say what this means for the drawing. Draw the 10 cm base. The arcs of radius 4 cm and 5 cm reach only 9 cm in total, so they never cross.The arcs do not meet, so there is no triangle. Katie is wrong.
Check. Try 4, 5, 8: 4 + 5 = 9 > 8, so that triangle exists. ✓
Watch out. The longest side must be shorter than the sum of the other two.
Key words. To construct means to draw accurately using only a ruler and a pair of compasses (no protractor). The construction arcs are the curved lines the compasses leave; you must not rub them out.
Method. Compare the sizes of the two triangles. Then use what you know about angles.
- Compare the shortest sides. Congruent triangles have exactly the same side lengths.5 cm ≠ 8 cm, so the triangles are not the same size
- Compare the angles. Both have angles 50°, 60°, 70°, so they have the same shape.Same angles means the same shape
- Write the conclusion. Triangles with the same shape but a different size are similar.They are similar but not congruent. AAA is not a condition for congruence.
Check. The scale factor is 8 ÷ 5 = 1.6. Every side of the larger triangle is 1.6 times the matching side of the smaller one. ✓
Watch out. Congruent means identical in shape and size.
Key words. To construct means to draw accurately using only a ruler and a pair of compasses (no protractor). The construction arcs are the curved lines the compasses leave; you must not rub them out.
Method. Construct and measure. Then calculate BC with the cosine rule a2 = b2 + c2 − 2bc cos A.
- Construct and measure. Use SAS as in the worked example.BC measures about 4.6 cm
- Write the cosine rule. BC is opposite angle A, so BC2 = AB2 + AC2 − 2 × AB × AC × cos A.BC2 = 62 + 4.52 − 2 × 6 × 4.5 × cos 50°
- Calculate. Evaluate each part.= 36 + 20.25 − 34.711 = 21.539
BC = √21.539 = 4.64 cm - Compare. Round the calculation to the nearest millimetre.4.64 cm = 4.6 cm to 1 d.p., which matches the measurement.
Check. A good drawing is accurate to 1 mm, which is 0.1 cm. ✓
Watch out. Use the angle between the two known sides in the cosine rule.
Key words. To construct means to draw accurately using only a ruler and a pair of compasses (no protractor). The construction arcs are the curved lines the compasses leave; you must not rub them out.
Method. Draw AB and the line from A at 30°. Point C is on that line and 4 cm from B, so it lies on an arc of radius 4 cm from B. The arc may cross the line at two points.
- Think about the distance from B to the line. B is 7 cm from A, and the line from A leans at 30°. The shortest distance from B to the line is the perpendicular distance.7 × sin 30° = 3.5 cm
- Compare with the arc radius. The arc of radius 4 cm reaches the line if 4 is more than 3.5 and the arc crosses twice if 4 is also less than 7.3.5 < 4 < 7, so the arc crosses the line at two points.
- Use the cosine rule to find AC. Let AC = x. Then BC2 = AB2 + x2 − 2 × AB × x cos 30°.16 = 49 + x2 − 14x cos 30°
x2 − 12.124x + 33 = 0 - Solve the quadratic equation. Use the formula.x = (12.124 ± √15.000) ÷ 2
x = 8.00 or x = 4.13 - Write the conclusion. Both values are positive and both triangles satisfy the given information.AC = 8.0 cm or AC = 4.1 cm. Two triangles are possible.
Check. Check x = 4.13: 4.132 + 49 − 14 × 4.13 × cos 30° = 16.00, close to 16. ✓
Watch out. Two sides and a non-included angle (SSA) does not always give a unique triangle.
Key words. To construct means to draw accurately using only a ruler and a pair of compasses (no protractor). The construction arcs are the curved lines the compasses leave; you must not rub them out.
Method. A rhombus is two congruent triangles. Construct a triangle with two sides of 5 cm and an angle of 60° between them (SAS), then repeat it on the other side of the diagonal.
- Construct the first triangle. SAS with AB = 5 cm, angle A = 60°, AD = 5 cm. This is an equilateral triangle.The short diagonal BD = 5 cm
- Complete the rhombus. All four sides are 5 cm, so draw arcs of radius 5 cm from B and D. They cross at C.BC = DC = 5 cm
- Measure the longer diagonal AC. It is the height of two equilateral triangles together.AC = 2 × 5 × cos 30° = 8.66 cm ≈ 8.7 cm
Check. Check with Pythagoras: half the short diagonal is 2.5 cm and half the long diagonal is 4.33 cm: 2.52 + 4.332 = 25.00 = 52. ✓
Watch out. The diagonals of a rhombus cross at 90° and cut each other in half.
Key words. To construct means to draw accurately using only a ruler and a pair of compasses (no protractor). The construction arcs are the curved lines the compasses leave; you must not rub them out.
Method. Share the perimeter in the ratio to find the sides, construct with SSS, then test for a right angle with Pythagoras.
- Find the value of one part. There are 3 + 4 + 5 = 12 parts in the ratio.24 ÷ 12 = 2 cm per part
- Find the sides. Multiply each ratio number by 2.3 × 2 = 6 cm, 4 × 2 = 8 cm, 5 × 2 = 10 cm
- Construct the triangle. Base 10 cm, arcs of radius 6 cm from one end and 8 cm from the other.The arcs cross at C.
- Test for a right angle. If 62 + 82 = 102 then the angle opposite the 10 cm side is 90°.62 + 82 = 36 + 64 = 100 = 102. The triangle is right-angled.
Check. Measure the angle between the 6 cm and 8 cm sides: it is 90°. ✓
Watch out. Use the longest side as the hypotenuse in Pythagoras.