IB Geometry worksheet on similar triangles word problems with diagrams and measurements.

Here are the step-by-step solutions for each problem on your worksheet. These problems all use similar triangles, which means the ratios of corresponding sides are equal.

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Problem 1

Given:
- Student’s eye height = 1.9 m
- Distance from student to mirror = 2.4 m
- Distance from mirror to statue = 4 m
- Find: Height of statue (let’s call it *h*)

Step-by-step:
The two right triangles formed (student-mirror and statue-mirror) are similar because the angle of incidence equals the angle of reflection.

So, we set up a proportion:

> (Height of statue) / (Distance from mirror to statue) = (Student’s height) / (Distance from student to mirror)

That is:

> h / 4 = 1.9 / 2.4

Now solve for *h*:

Multiply both sides by 4:

> h = (1.9 / 2.4) × 4

Calculate 1.9 ÷ 2.4 first:

> 1.9 ÷ 2.4 ≈ 0.791666...

Then multiply by 4:

> 0.791666... × 4 ≈ 3.166666...

Which is approximately 3.17 meters (rounded to two decimal places).

But let’s do it exactly as fractions:

1.9 = 19/10
2.4 = 24/10 = 12/5

So:

> h = (19/10) ÷ (12/5) × 4
= (19/10) × (5/12) × 4
= (19 × 5 × 4) / (10 × 12)
= (380) / (120)
= 38/12
= 19/6
≈ 3.1667 m

✔ So, exact value is 19/6 meters, or about 3.17 m

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Problem 2

Given:
- Girl stands 15 m away from statue → so distance from girl to mirror + mirror to statue = 15 m
- Mirror is placed 12 m from statue → so mirror is 3 m from girl (since 15 - 12 = 3)
- Girl’s height = 5.5 ft
- Find: Statue’s height (*H*)

Again, similar triangles:

> H / 12 = 5.5 / 3

Solve for H:

> H = (5.5 / 3) × 12
= 5.5 × 4
= 22 feet

✔ Final Answer: 22 ft

*(Note: Units are mixed — meters for distances, feet for heights — but since the ratio uses same units for height and same for distance, it cancels out. We’re told girl’s height is in ft, so answer is in ft.)*

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Problem 3

Given:
- Tree height = 24 ft
- Tree shadow = 12 ft
- Brad’s height = 6 ft
- Find: Brad’s shadow length (*s*)

Similar triangles again: sun’s rays create similar triangles for tree and Brad.

> Tree height / Tree shadow = Brad’s height / Brad’s shadow
> 24 / 12 = 6 / s

Simplify left side:

> 2 = 6 / s

Multiply both sides by s:

> 2s = 6
→ s = 3

✔ Final Answer: 3 feet

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Problem 4

Given:
- Flagpole height = 40 ft
- Flagpole shadow = 25 ft
- Building height = 200 ft
- Find: Building’s shadow (*S*)

Set up proportion:

> Flagpole height / Flagpole shadow = Building height / Building shadow
> 40 / 25 = 200 / S

Cross-multiply:

> 40 × S = 25 × 200
> 40S = 5000
> S = 5000 ÷ 40
> S = 125

✔ Final Answer: 125 feet

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Problem 5

Given:
- Girl’s height = 160 cm
- Distance from lamp post = 360 cm
- Shadow length = 90 cm
- Find: Lamp post height (*L*)

This one is trickier — the shadow is cast *from the light*, so the total base of the big triangle is:

> Distance from lamp post to end of shadow = 360 cm + 90 cm = 450 cm

The small triangle is the girl and her shadow: height 160 cm, base 90 cm.

The large triangle is the lamp post and the full shadow: height L, base 450 cm.

They are similar triangles!

So:

> Girl’s height / Her shadow = Lamp post height / Total shadow length
> 160 / 90 = L / 450

Solve for L:

> L = (160 / 90) × 450
= (16/9) × 450
= 16 × 50
= 800 cm

✔ Final Answer: 800 cm

*(You can also think: 450 ÷ 90 = 5, so scale factor is 5 → 160 × 5 = 800)*

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## ✔ Final Answers:

1. 19/6 meters or approximately 3.17 meters
2. 22 feet
3. 3 feet
4. 125 feet
5. 800 cm