Similar Triangles & Proportions with Shadows

  • 10th Grade
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| Attempts: 12 | Questions: 20 | Updated: Dec 11, 2025
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1) A 10 ft post casts a 7 ft shadow. At the same time, a statue casts a 9.8 ft shadow. What is the statue’s height (nearest tenth of a foot)?

Explanation

(height / shadow) constant ⇒ 10 / 7 = x / 9.8

x = 9.8 · (10/7) = 14.0 ft

Hence, 14.0 ft.

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About This Quiz
Similar Triangles & Proportions With Shadows - Quiz

Use proportional reasoning and similar triangles to link different objects casting shadows at the same time. You’ll scale heights and shadows, translate “same sun angle” into constant ratios, and decide when to use tan(θ) vs. direct proportions. Expect quick computations, ratio checks, and interpreting how changing the sun angle affects... see moreshadow length.
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2) A mural wall of unknown height H casts a shadow S. If at the same time a 2.0 m stick casts a 0.8 m shadow, which equation relates H and S?

Explanation

H/S = height/shadow (same angle) = 2.0 / 0.8 = 2.5

Hence, H/S = 2.0/0.8.

Submit
3) When the sun’s elevation is 30°, a 9 m tower’s shadow is L. When the elevation becomes 60°, what is the new shadow length in terms of L?

Explanation

L₁ = 9 / tan30° = 9 / (√3/3) = 9√3

L₂ = 9 / tan60° = 9 / √3 = 3√3

L₂ / L₁ = (3√3) / (9√3) = 1/3 ⇒ L₂ = L/3

Hence, L/3.

Submit
4) A 2.5 m sign casts a 1.0 m shadow. A nearby tower casts a 16.0 m shadow. What is the tower’s height?

Explanation

Ratio = height/shadow = 2.5 / 1.0 = 2.5

Tower height = 2.5 · 16.0 = 40 m

Hence, 40 m.

Submit
5) A 1.6 m pole casts a 2.0 m shadow. At the same time, a tree's shadow is 7.5 m. What is the tree's height (nearest tenth of a meter)?

Explanation

(height / shadow) constant ⇒ 1.6 / 2.0 = 0.8

Tree height = 0.8 · 7.5 = 6.0 m

Hence, 6.0 m.

Submit
6) The sun’s elevation is θ. Object A has height 3 m and shadow 2 m. Which expression gives the shadow of Object B with height 4.5 m?

Explanation

tan(θ) = 3 / 2 ⇒ shadow = height / tan(θ)

Shadow_B = 4.5 / (3/2) = 4.5 · (2/3) = 3

Hence, 3.

Submit
7) Two towers stand on level ground when the sun’s elevation is 35°. Tower T1 is 24 m tall. Which equation gives T1’s shadow length s1?

Explanation

tan(35°) = 24 / s1 ⇒ s1 = 24 / tan(35°)

Hence, option C.

Submit
8) A sign 2.4 m tall casts a 1.5 m shadow. A nearby building casts a 17.5 m shadow at the same time. What is the building’s height?

Explanation

Ratio = 2.4 / 1.5 = 1.6

Building height = 1.6 · 17.5 = 28.0 m

Hence, 28.0 m.

Submit
9) On level ground, Object A: height 5 m, shadow 4 m. Object B has shadow 10 m at the same time. What is Object B’s height?

Explanation

Ratio = 5 / 4 = 1.25

Height_B = 1.25 · 10 = 12.5 m

Hence, 12.5 m.

Submit
10) A 3.2 m pole casts a 1.0 m shadow at noon. Later, the same pole casts a 2.0 m shadow. Which statement is true about the later sun angle?

Explanation

Longer shadow ⇒ smaller elevation angle.

Hence, decreased (sun lower).

Submit
11) The sun’s elevation is 50°. A 6 m flagpole casts a shadow L1. Which is the shadow L2 of a 9 m flagpole at the same time?

Explanation

Shadows ∝ heights (same angle).

9 m is 1.5 × 6 m ⇒ L2 = 1.5 L1

Hence, 1.5 L1.

Submit
12) A lamppost 4.5 m tall casts a 3.0 m shadow. Another object has height 7.5 m. What is its shadow (nearest tenth of a meter)?

Explanation

Ratio = 4.5 / 3.0 = 1.5

Shadow = 7.5 / 1.5 = 5.0 m

Hence, 5.0 m.

Submit
13) Two trees, A and B, stand on level ground. A is 8 m tall with a 5 m shadow. B has a 12.5 m shadow. What is B’s height?

Explanation

Ratio = 8 / 5 = 1.6

Height_B = 1.6 · 12.5 = 20.0 m

Hence, 20.0 m.

Submit
14) A 1.5 m post casts a shadow of length s1 at sun angle θ. A 4.5 m pole casts a shadow s2 at the same time. Which relation is correct?

Explanation

Shadows ∝ heights ⇒ s2 / s1 = 4.5 / 1.5 = 3

Hence, s2 = 3 s1.

Submit
15) A 2.0 m bollard casts a 1.2 m shadow. A sculpture is 3.5 m tall. What is the sculpture’s shadow length (nearest tenth of a meter)?

Explanation

Ratio = 2.0 / 1.2 = 1.666…

Shadow = 3.5 / 1.666… = 3.5 · (3/5) = 2.1 m

Nearest tenth: 2.1 m → Option B

Hence, 2.1 m.

Submit
16) On a certain day, tan(θ) = 0.8. An object’s height is H and its shadow is L. Which statement is true?

Explanation

tan(θ) = height / shadow = H / L = 0.8

Hence, H/L = 0.8.

Submit
17) A 12 ft post casts a shadow of 9 ft. At the same time, another post has height 18 ft. What is its shadow (nearest tenth of a foot)?

Explanation

Ratio = 12 / 9 = 4/3

Shadow = 18 / (4/3) = 18 · (3/4) = 13.5 ft

Hence, 13.5 ft.

Submit
18) Two vertical poles cast shadows on level ground at the same time. Pole A: height 7 m, shadow 4 m. Pole B: height 10.5 m, shadow x. Find x.

Explanation

x / 4 = 10.5 / 7 = 1.5

x = 4 · 1.5 = 6.0 m

Hence, 6.0 m.

Submit
19) The sun’s elevation is θ. Object X has height 5 m and shadow 3 m. Which equation expresses tan(θ)?

Explanation

tan(θ) = height / shadow = 5 / 3

Hence, 5/3.

Submit
20) A 20 m mast casts a 12 m shadow. Later, the shadow is 15 m. What happened to the sun’s elevation?

Explanation

Longer shadow ⇒ smaller elevation angle.

Hence, it decreased.

Submit
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A 10 ft post casts a 7 ft shadow. At the same time, a statue casts a...
A mural wall of unknown height H casts a shadow S. If at the same time...
When the sun’s elevation is 30°, a 9 m tower’s shadow is L. When...
A 2.5 m sign casts a 1.0 m shadow. A nearby tower casts a 16.0 m...
A 1.6 m pole casts a 2.0 m shadow. At the same time, a tree's shadow...
The sun’s elevation is θ. Object A has height 3 m and shadow 2 m....
Two towers stand on level ground when the sun’s elevation is 35°....
A sign 2.4 m tall casts a 1.5 m shadow. A nearby building casts a 17.5...
On level ground, Object A: height 5 m, shadow 4 m. Object B has shadow...
A 3.2 m pole casts a 1.0 m shadow at noon. Later, the same pole casts...
The sun’s elevation is 50°. A 6 m flagpole casts a shadow L1. Which...
A lamppost 4.5 m tall casts a 3.0 m shadow. Another object has height...
Two trees, A and B, stand on level ground. A is 8 m tall with a 5 m...
A 1.5 m post casts a shadow of length s1 at sun angle θ. A 4.5 m pole...
A 2.0 m bollard casts a 1.2 m shadow. A sculpture is 3.5 m tall. What...
On a certain day, tan(θ) = 0.8. An object’s height is H and its...
A 12 ft post casts a shadow of 9 ft. At the same time, another post...
Two vertical poles cast shadows on level ground at the same time. Pole...
The sun’s elevation is θ. Object X has height 5 m and shadow 3 m....
A 20 m mast casts a 12 m shadow. Later, the shadow is 15 m. What...
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