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You cannot measure the height of a tree directly, but you can measure its shadow and the shadow of a post beside it. The two triangles have the same shape, so their sides are in the same ratio. This lesson shows how to prove two triangles similar, match their sides correctly, and use the scale factor to find unknown lengths.
Equal angles fix the matching
Similar triangles have the same shape. Their corresponding angles are equal, and their corresponding sides are in the same ratio. That common ratio is the scale factor.
Match vertices by angles
One arc marks the equal angles at A and P. Two arcs mark the equal angles at B and Q. Triangle PQR is reflected, so Q is on the right and R on the left.
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Figure data
This table is included in print while it is open.
| Item | Value |
|---|---|
| Equal angles | ∠A = ∠P, ∠B = ∠Q |
| AB | 6 cm |
| BC | 8 cm |
| AC | 5 cm |
| PQ | 9 cm |
Two pairs of equal angles are enough. The third angles are equal too, because the angles of each triangle add up to 180°. So A matches P, B matches Q and C matches R. Write the triangles in that order: triangle ABC is similar to triangle PQR.
The vertex order tells you which sides match. AB matches PQ, BC matches QR and AC matches PR:
Triangle PQR is drawn the other way round. Match the vertices by their angles, not by where they sit on the page.
Worked example 1
Find the factor from one complete pair
Worked example 1
Find the factor from one complete pair
Use the diagram. Find the scale factor from triangle ABC to triangle PQR. Then find QR and PR.
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Choose a pair with both lengths known
Method
Divide the length in PQR by the matching length in ABC.Reason
AB joins the angles at A and B. PQ joins the matching angles at P and Q, so these two sides correspond.Working
Answer
scale factor = PQ/(AB) = 9/6 = 1.5.Multiply each matching side
Method
Multiply BC and AC by 1.5.Reason
QR matches BC and PR matches AC. Every length in PQR is 1.5 times its partner in ABC.Working
Answer
QR = 1.5 × 8 = 12 cm,; PR = 1.5 × 5 = 7.5 cm.
Quick check
All three ratios agree: 9/6 = 12/8 = 7.5/5 = 1.5. The scale factor has no unit, because it compares two lengths in the same unit.
Check your understanding 1
Use a factor smaller than 1
Check your understanding 1
Use a factor smaller than 1
Triangle XYZ is similar to triangle ABC above, with X, Y and Z matching A, B and C. In triangle XYZ, YZ = 6 cm. Find the scale factor from triangle ABC to triangle XYZ, then find XY and XZ.
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YZ matches BC, so the scale factor is 6/8 = 0.75. A factor less than 1 makes a smaller triangle. Then XY = 0.75 × 6 = 4.5 cm and XZ = 0.75 × 5 = 3.75 cm.
A parallel line cuts off a similar triangle
In triangle ABC, D is on AB and E is on AC, with DE ∥ BC. The arrows on DE and BC show that they are parallel.
Small triangle ADE inside triangle ABC
The small triangle ADE shares vertex A with the large triangle ABC. Matching arrows on DE and BC show that the lines are parallel.
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Figure data
This table is included in print while it is open.
| Item | Value |
|---|---|
| Parallel lines | DE ∥ BC |
| AD | 4 cm |
| DB | 6 cm |
| AE | 3.2 cm |
| DE | 5 cm |
Triangles ADE and ABC share the angle at A. Because DE ∥ BC, corresponding angles give ∠ ADE = ∠ ABC and ∠ AED = ∠ ACB. So triangle ADE is similar to triangle ABC, with D matching B and E matching C.
The large triangle’s sides are whole sides. AD matches AB, and AB = AD + DB = 4 + 6 = 10 cm. DB is only part of a side, so it belongs to neither triangle.
Worked example 2
Find a whole side and a part of a side
Worked example 2
Find a whole side and a part of a side
Use the diagram. Find BC and EC.
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Find the factor from whole sides
Method
Divide AB by AD.Reason
AD in the small triangle matches AB in the large triangle. Both are complete sides of their triangles.Working
Answer
scale factor from ADE to ABC = AB/(AD) = 10/4 = 2.5.Scale the matching sides
Method
Multiply DE and AE by 2.5. Then subtract AE from AC.Reason
BC matches DE. EC is not a side of either triangle, so find the whole side AC first.Working
Answer
BC = 2.5 × 5 = 12.5 cm,; AC = 2.5 × 3.2 = 8 cm,; EC = AC - AE = 8 - 3.2 = 4.8 cm.
Quick check
Check the whole sides: AB/(AD) = 10/4, AC/(AE) = 8/3.2 and BC/(DE) = 12.5/5 all equal 2.5. In the other direction, the factor from ABC to ADE is 4/10 = 0.4.
Check your understanding 2
Find the error
Check your understanding 2
Find the error
Using the same diagram, a learner writes BC/(DE) = DB/(AD) and gets BC = 6/4 × 5 = 7.5 cm. Explain the mistake. Then give the scale factor from triangle ABC to triangle ADE.
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DB is part of side AB, so it is not a side of triangle ADE or triangle ABC. AD matches the whole side AB = 10 cm, so the correct factor from ADE to ABC is 2.5 and BC = 12.5 cm. From ABC to ADE, the factor is the reciprocal: 4/10 = 0.4.
Crossed lines turn the second triangle over
Lines AC and BD cross at O, and AB ∥ DC.
Two triangles meeting at O
Triangle OAB is above O and triangle OCD is below O. A and C are at opposite ends of one straight line, and B and D are at opposite ends of the other.
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Figure data
This table is included in print while it is open.
| Item | Value |
|---|---|
| Crossing lines | AC and BD meet at O |
| Parallel lines | AB ∥ DC |
| OA | 6 cm |
| OB | 4 cm |
| AB | 8 cm |
| OC | 9 cm |
The angles at O are vertically opposite, so ∠ AOB = ∠ COD. Because AB ∥ DC, alternate angles give ∠ OAB = ∠ OCD and ∠ OBA = ∠ ODC. So triangle OAB is similar to triangle OCD, with A matching C and B matching D. D sits below A in the drawing, but it does not match A.
Check your understanding 3
Match across the crossing
Check your understanding 3
Match across the crossing
Find the scale factor from triangle OAB to triangle OCD. Then find CD and OD.
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A matches C, so OA matches OC and the factor is 9/6 = 1.5. CD matches AB, so CD = 1.5 × 8 = 12 cm. OD matches OB, so OD = 1.5 × 4 = 6 cm.
Measure heights with similar triangles
At the same moment, the sun’s rays are parallel. On level ground they make the same angle with the ground. A vertical post and its shadow form a right-angled triangle. A vertical person and their shadow form another. Two pairs of equal angles make these triangles similar.
Two shadows at the same time
Both uprights stand at right angles to level ground. The angle between each dashed ray and the ground is marked.
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Figure data
This table is included in print while it is open.
| Item | Value |
|---|---|
| Person's height | 1.6 m |
| Person's shadow | 2.4 m |
| Pole's shadow | 15 m |
| Pole's height | h m |
Worked example 3
Find the height of the pole
Worked example 3
Find the height of the pole
Find the height of the pole.
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Match the sides
Method
Pair height with height and shadow with shadow.Reason
Each height is opposite the sun’s angle. Each shadow is beside the sun’s angle and the right angle.Working
Answer
scale factor from person to pole = 15/2.4 = 6.25.Scale the height
Method
Multiply the person’s height by 6.25.Reason
The pole’s height matches the person’s height.Working
Answer
h = 6.25 × 1.6 = 10.The pole is 10 m tall.
Quick check
You can also compare height with shadow inside each triangle: 1.6/2.4 = h/15 gives the same h = 10. This model assumes level ground and vertical uprights.
Check your understanding 4
Support a ramp
Check your understanding 4
Support a ramp
A straight ramp rises 0.9 m over a horizontal distance of 6 m. A vertical post supports the ramp 2 m horizontally from its lower end. Sketch the ramp and post. Name the two similar triangles, then find the height of the post.
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The post, the ramp and the ground form a small right-angled triangle. It shares the angle at the lower end with the whole ramp triangle, so the triangles are similar. The 2 m horizontal side matches the 6 m horizontal side, so the factor is 2/6 = 1/3. The post matches the 0.9 m rise: 1/3 × 0.9 = 0.3 m.
One equal angle is not enough
Check your understanding 5
Check the information
Check your understanding 5
Check the information
Triangles KLM and XYZ both have a 50° angle, at K and at X. KL = 4 cm, XY = 6 cm and LM = 5 cm. A learner says that YZ must be 7.5 cm. Is that justified?
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No. One pair of equal angles does not fix the shape. The other angles of the two triangles could be different, so the triangles need not be similar and YZ is not determined. If you also knew that ∠ L = ∠ Y, the third angles would be equal too. Then KL would match XY and LM would match YZ, giving YZ = 6/4 × 5 = 7.5 cm.
Common mistakes
- Matching sides by position on the page. Match vertices by equal angles, then list the triangles in that order.
- Using part of a side. In a triangle cut by a parallel line, use whole sides such as AB = AD + DB.
- Inverting the scale factor. Divide a length in the new triangle by the matching length in the original.
- Claiming similarity from one equal angle. Two pairs of equal angles, or all three sides in the same ratio, are needed.
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