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Trigonometry · Trigonometry: Similar Triangles, Right Triangles and Acute Triangles

Finding Unknown Lengths in Similar Triangles

1.1 1.3About 70 minutes
Learning goals
  • I can set up a proportion using corresponding sides of similar triangles.
  • I can solve a proportion for an unknown length.
  • I can find unknown lengths in nested (parallel-line) and "bow tie" diagrams.
  • I can check that an answer is reasonable using the scale factor.

Words to know

proportion
An equation stating that two ratios are equal, such as x/6 = 10/4.
cross-multiplying
Rewriting a/b = c/d as ad = bc to clear the fractions.
nested triangles
Two triangles that share a vertex, where one sits inside the other (for example, created by a line parallel to one side).
corresponding sides
Sides in matching positions of similar figures: they sit opposite equal angles.

Similar triangles as a measuring tool

Once you know two triangles are similar, every pair of corresponding sides has the same ratio. If you know three of the lengths in two pairs, you can find the fourth without measuring it. Builders, surveyors and designers use this every day: you will see it in ramps, roof frames and bridges.

A five-step method

  1. Show the triangles are similar (AA, SSS or SAS) and write the similarity statement in matching order.
  2. Redraw the triangles separately, both facing the same way, and label every known length.
  3. Write a proportion of corresponding sides. Put the unknown in a numerator if you can.
  4. Solve: multiply both sides by the denominator under the unknown, or cross-multiply.
  5. Check: the bigger triangle must have the bigger sides, and every pair of sides must give the same scale factor.
Worked example

Example 1: A direct proportion

△ABC ~ △DEF. AB = 6 cm, BC = 9 cm and DE = 10 cm. Find EF.

  1. Matching letters: AB ↔ DE and BC ↔ EF.
  2. Proportion with the unknown on top: EF/BC = DE/AB, so EF/9 = 10/6.
  3. Multiply both sides by 9: EF = 9 × 10 ÷ 6 = 15.
  4. Check: k = 10 ÷ 6 ≈ 1.67, and 15 ÷ 9 ≈ 1.67. ✓

EF = 15 cm

Worked example

Example 2: Nested triangles (watch the whole side!)

In △ABC, DE ∥ BC with D on AB and E on AC. AD = 4, DB = 6 and DE = 5. Find BC.

  1. DE ∥ BC, so △ADE ~ △ABC (AA: shared ∠A and corresponding angles).
  2. The side of the big triangle is the whole side: AB = AD + DB = 4 + 6 = 10.
  3. Corresponding sides: AD ↔ AB and DE ↔ BC. Proportion: BC/DE = AB/AD, so BC/5 = 10/4.
  4. BC = 5 × 10 ÷ 4 = 12.5.
  5. Common mistake: using DB = 6 instead of AB = 10 gives 7.5, which is wrong. DB is not a side of either triangle.

BC = 12.5

ABCDE465BC = ?A5DE4A?BC10AB = 4 + 6 = 10
Left: the nested diagram for Example 2. Right: the same two triangles pulled apart and redrawn. Redrawing makes it obvious that the big triangle's side is AB = 10, not 6.
Worked example

Example 3: When the unknown is part of a side

In △ABC, DE ∥ BC. AD = x, DB = 3, DE = 4 and BC = 6. Find x.

  1. △ADE ~ △ABC, so AD/AB = DE/BC, where AB = x + 3.
  2. x/(x + 3) = 4/6.
  3. Cross-multiply: 6x = 4(x + 3), so 6x = 4x + 12.
  4. 2x = 12, so x = 6.
  5. Check: AD/AB = 6/9 = 2/3 and DE/BC = 4/6 = 2/3. ✓

x = 6, so AD = 6.

Worked example

Example 4: A bow tie

Segments AE and BD cross at C, and AB ∥ DE. AC = 6, CE = 9 and AB = 8. Find DE.

  1. From Lesson 2: △ABC ~ △EDC (vertically opposite angles at C, alternate angles at A and E).
  2. Corresponding sides: AC ↔ EC and AB ↔ ED.
  3. DE/AB = EC/AC, so DE/8 = 9/6.
  4. DE = 8 × 9 ÷ 6 = 12.

DE = 12

Worked example

Example 5: Building a ramp

A straight ramp starts on the ground and rises to a deck. Its whole length, measured along the ground, is 5.0 m. A vertical support post 2.0 m from the bottom end is 0.4 m tall. How high is the top of the ramp?

  1. The post and the deck edge are both vertical, so each makes a right angle with the ground. The angle at the bottom of the ramp is shared. So the small triangle and the big triangle are similar (AA).
  2. Proportion: h/0.4 = 5.0/2.0.
  3. h = 0.4 × 5.0 ÷ 2.0 = 1.0.

The top of the ramp is 1.0 m high.

0.4 mh = ?2.0 m5.0 m (whole ramp, measured along the ground)ramp surface
The support post cuts off a small right triangle that is similar to the whole ramp triangle. Here the scale factor is 5.0 ÷ 2.0 = 2.5.

Solving proportions quickly

  • If the unknown is on top: x/9 = 10/6 → multiply both sides by 9.
  • If the unknown is on the bottom: 15/x = 5/8 → cross-multiply: 5x = 120, so x = 24.
  • If the unknown appears in a sum (like x + 3): cross-multiply, expand the brackets, then collect the x-terms, exactly like the equations you solved in Grade 9.
Just for fun

Why are similar triangles never jealous of each other?

They know they're all in proportion. Nobody gets more than their fair share.

Hands-on activity

Book-ramp check

You need: a long straight board, metre stick or a 30 cm ruler, a stack of books, and a measuring tape or second ruler.

  1. Rest one end of the board on the floor or table and the other on the stack of books to make a ramp.
  2. Measure the horizontal distance from the bottom end of the ramp to the books, and the height of the top. Write height ÷ horizontal distance.
  3. Choose a point partway up the ramp. Measure straight down from the ramp to the table (keep your ruler vertical), and the horizontal distance from the bottom end to that point. Find this ratio too.
  4. Repeat at one more point.

What to look for: all three ratios should be almost equal. Each measurement makes a right triangle with the same bottom angle, so the triangles are similar. Use your ratio to predict the height at another point before you measure it.

Your turn

1

△PQR ~ △STU with PQ = 8, QR = 12 and ST = 6. Find TU.

2

In the same triangles, SU = 7.5. Find PR.

3

In △ABC, DE ∥ BC, AD = 3, AB = 12 and DE = 2. Find BC.

4

In △ABC, DE ∥ BC, AD = 5, DB = 10 and AE = 4. Find EC.

5

AE and BD cross at C with AB ∥ DE (△ABC ~ △EDC). AC = 4, CE = 10 and BC = 5. Find CD.

6

In △ABC, DE ∥ BC, AD = x, DB = 4, DE = 6 and BC = 9. Find x.

7

Solve 15/x = 5/8.

More practice: warm-up

1

Solve x/4 = 9/6.

2

Solve 10/x = 2/3.

3

Two triangles are similar with scale factor 2.5. A side of the smaller one is 4 cm. Find the matching side of the larger one.

4

A triangle is reduced with scale factor 0.4. A side was 20 cm. Find the new length.

5

△ABC ~ △DEF. Name a ratio that equals AB/DE.

More practice: on your own

1

A triangular wall frame has a horizontal brace DE parallel to the floor beam BC. The sloped side AB is 3.6 m, AD = 1.2 m (from the peak A to the brace), and BC = 4.5 m. How long is the brace?

2

AE and BD cross at C with AB ∥ DE. AC = x, CE = 12, AB = 5 and DE = 15. Find x.

3

In △ABC, DE ∥ BC, AD = 6, DB = 9 and BC = 20. Find DE.

4

In right △ABC (∠C = 90°), CD ⊥ AB, so △ADC ~ △ACB (Lesson 2). AC = 10 and AB = 12.5. Find AD.

5

Sam's work: "DE ∥ BC, AD = 4, DB = 6, BC = 15. DE/BC = AD/DB, so DE = 15 × 4 ÷ 6 = 10." Find the error and the correct answer.

Challenge

1

In △ABC, DE ∥ BC. AD = x, DB = x + 2, DE = 5 and BC = 15. Find x and AB.

2

In △ABC, segments DE and FG are both parallel to BC, with D and F on AB. AD = 2, DF = 3, FB = 5 and BC = 20. Find DE and FG.

3

In Grade 9 you saw that multiplying every length by k multiplies area by k². △ADE ~ △ABC with AD = 3 and AB = 9. The area of △ABC is 54 cm². Find the area of △ADE.

Check your understanding

Quiz

1.

△ABC ~ △DEF with AB = 4, DE = 6 and BC = 10. What is EF?

2.

In △ABC, DE ∥ BC, AD = 2, DB = 6 and DE = 3. What is BC?

3.

Solve x/5 = 12/15.

4.

In a nested-triangle diagram with D on AB, which length is a side of the large triangle?

5.

In a bow tie with AB ∥ DE (△ABC ~ △EDC), AC = 3, CE = 6 and AB = 4. What is DE?

6.

A ramp is 6 m long (measured along the ground). A 0.5 m vertical post stands 2 m from the bottom end. How high is the top?

For parents and teachers

Big idea: Using similar triangles to find lengths (expectation 1.1) and beginning to apply this in realistic settings (1.3).

How to help: Ask your teen to redraw the two triangles separately before writing any proportion; this one habit prevents most errors. Then ask "Which side matches which?" and "Does your answer make the big triangle bigger?"

Common difficulties:

  • Using a part of a side (DB) instead of the whole side (AB) in nested triangles.
  • Mixing up the order in a proportion (small/big on one side, big/small on the other).
  • Algebra slips when the unknown is in a sum, such as x/(x + 3). Review expanding brackets if needed.
  • Forgetting to subtract at the end when the question asks for a part (EC = AC − AE).