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

Similar Triangles in the Real World: Shadows, Mirrors, Models and Surveying

1.3About 70 minutes

Learning goals

I can use shadows and similar triangles to find the height of a tall object.
I can use the mirror method to find a height I cannot reach.
I can use scale factors to solve problems with scale models and scale drawings.
I can use similar triangles to find a distance I cannot measure directly, such as the width of a river.

Words to know

indirect measurement
Finding a length by calculation instead of measuring it directly.

scale
The ratio of a length on a model or drawing to the real length, such as 1:50.

law of reflection
Light bounces off a mirror at the same angle it hits it.

surveying
Measuring land: distances, heights and angles, often using triangles.

Measuring the unmeasurable

How tall is the maple tree in the park? How wide is the river? You could climb the tree or swim the river with a tape measure... or you could use similar triangles. This is called indirect measurement, and people have used it for thousands of years.

Method 1: Shadows

The sun is so far away that its rays arrive almost parallel. At one moment, every vertical object and its shadow form a right triangle, and the sun's rays meet the ground at the same angle in all of them. Two angles match (the right angle and the sun angle), so the triangles are similar by AA.

Safety: never look directly at the sun, even through sunglasses. You only need to look at the shadows on the ground.

A person and a tree at the same moment. The dashed sun rays are parallel, so the two shaded right triangles are similar. (Each side is drawn at its own scale.)

Worked example

Example 1: How tall is the tree?

At 2 p.m. a 1.6 m tall student casts a shadow 2.0 m long. At the same time a tree casts a shadow 15 m long. How tall is the tree?

The two right triangles are similar (right angle and same sun angle).
Height ↔ height and shadow ↔ shadow: h/1.6 = 15/2.0.
h = 1.6 × 15 ÷ 2.0 = 12.

The tree is about 12 m tall.

Method 2: A mirror on the ground

Light bounces off a mirror at the same angle it arrives (the law of reflection). Put a small mirror flat on the ground between you and a flagpole. Step back until you can see the top of the pole in the mirror. Then:

the angle between the ground and the light from the pole top equals the angle between the ground and the light going to your eye, and
both you and the pole stand at right angles to the ground.

AA again: the two triangles are similar.

The mirror method. The light from the top of the pole reflects to the student's eye, making equal angles with the ground. Use eye height, not full height.

Worked example

Example 2: The mirror method

Maya's eyes are 1.5 m above the ground. She stands 2.0 m from a mirror and sees the top of a flagpole in it. The mirror is 8.0 m from the base of the pole. How tall is the pole?

Small triangle: eye height 1.5 m, distance 2.0 m. Large triangle: pole height h, distance 8.0 m.
h/1.5 = 8.0/2.0.
h = 1.5 × 4 = 6.0.

The flagpole is about 6.0 m tall.

Method 3: Scale models and drawings

A scale like 1:24 means 1 unit on the model stands for 24 of the same units in real life. The model and the real object are similar, with scale factor 24 (from model to real) or 1/24 (from real to model).

Real length = model length × 24
Model length = real length ÷ 24

Always use the same units on both sides of the scale.

Worked example

Example 3: A model car

A model car is built at a scale of 1:24. The model is 19 cm long. How long is the real car, in metres?

Real length = 19 × 24 = 456 cm.
456 cm ÷ 100 = 4.56 m.

The real car is 4.56 m long.

Method 4: Surveying across a river

You cannot stretch a tape measure across a river, but you can measure along your own side. Pick a tree A on the far bank. Stand at B, directly across from it. Then:

Walk along the bank 40 m to C and put in a stake.
Keep walking 10 m more to D.
Turn 90° away from the river and walk until the stake at C lines up with the tree. Call this point E. Measure DE = 7.5 m.

△ABC and △EDC both have a right angle, and the angles at C are vertically opposite, so △ABC ~ △EDC (AA).

Surveying a river. A, C and E are in a straight line, so the angles at C are equal. The two right triangles are similar. (Drawn to scale.)

Worked example

Example 4: How wide is the river?

Use the measurements in the diagram (BC = 40 m, CD = 10 m, DE = 7.5 m) to find the width AB.

Corresponding sides: AB ↔ ED and BC ↔ DC.
AB/ED = BC/DC, so AB/7.5 = 40/10.
AB = 7.5 × 4 = 30.

The river is about 30 m wide.

Just for fun

An ancient Greek story says that Thales of Miletus worked out the height of the Great Pyramid of Egypt using its shadow and the shadow of a stick. No ladder, no climbing: just similar triangles.

Hands-on activity

Shadow survey (sunny day)

You need: a sunny day, a measuring tape, a partner, a notebook. Never look directly at the sun.

Measure your partner's height.
Have your partner stand on flat ground with their back to the sun. Measure from their heels to the tip of the shadow of their head.
Right away (the sun moves!), measure the shadow of something tall: a lamp post, a basketball net post, a tree or a fence post. Measure from its base to the shadow's tip.
Set up height/shadow = height/shadow and solve for the tall object's height.
If you can measure the object directly (a fence post or a hoop post), compare your answer.

What to look for: a correct proportion with heights together and shadows together. Answers within about 10% of the real height are good. Shadows of trees are harder because the top is fuzzy, so expect more error. Bonus: redo it an hour later. The shadows change, but your answer should not.

Your turn

A 2.0 m post casts a 3.0 m shadow. At the same time, a building casts a 24 m shadow. How tall is the building?

Jordan is 1.8 m tall and casts a 1.2 m shadow. A tree beside him is 9.6 m tall. How long is the tree's shadow?

Leah's eyes are 1.6 m above the ground. She stands 1.2 m from a mirror and sees the top of a tree. The mirror is 9.0 m from the tree. How tall is the tree?

A model house is built at a scale of 1:50. The model is 24 cm tall. How tall is the real house, in metres?

A real canoe is 4.8 m long. How long is a 1:20 model of it, in centimetres?

In a river survey like Example 4, BC = 50 m, CD = 20 m and DE = 12 m. How wide is the river?

Why must both shadows be measured at about the same time?

More practice: warm-up

On a map with scale 1:50 000, two towns are 4 cm apart. How far apart are they in kilometres?

A 30 m bridge is modelled at 1:100. How long is the model in centimetres?

A metre stick (1 m) casts a 1.5 m shadow. A pole casts a 6 m shadow. How tall is the pole?

In the mirror method, which two angles are equal because of the law of reflection?

Name two real situations where similar triangles help measure something.

More practice: on your own

Ali holds a ruler upright at arm's length, 60 cm from his eye. A building 80 m away looks exactly 4.5 cm tall on the ruler. How tall is the building?

To find the length of a pond AB, Ethan stands at a point C where he can see both ends. He marks D on line AC beyond C with CD = ½ AC, and E on line BC beyond C with CE = ½ BC. He measures DE = 45 m. Explain why △ABC ~ △DEC and find AB.

A tree's shadow is 7.5 m long. At the same time, 1.5 m tall Lisa casts a 1.25 m shadow. How tall is the tree?

A floor plan has a scale of 1 cm : 0.5 m. A bedroom on the plan is 4.2 cm by 6.0 cm. What are the real dimensions?

A flagpole is known to be 7.5 m tall. Ben's eyes are 1.5 m high and he stands 1.8 m from a mirror on the ground when he sees the top of the pole. How far is the mirror from the pole?

Challenge

Priya (1.5 m tall) stands 12 m from a tree, between the tree and the tip of its shadow. The tip of her shadow lands exactly on the tip of the tree's shadow. Her shadow is 3 m long. How tall is the tree?

A pinhole camera is a box with a tiny hole. A 2 m tall person stands 5 m in front of the hole. The back of the box is 20 cm behind the hole. How tall is the (upside-down) image on the back of the box?

A 1:50 scale model of a deck has an area of 24 cm². What is the area of the real deck in square metres? (Hint: Grade 9 showed that area scales by k².)

Check your understanding

Quiz

A 1 m stick casts a 0.8 m shadow. A flagpole casts a 6.4 m shadow at the same time. How tall is the flagpole?

Why are the two triangles in a shadow problem similar?

A 1:25 model is 8 cm tall. How tall is the real object?

Eyes 1.5 m high, 2 m from a mirror; the mirror is 10 m from a building. How tall is the building?

In the river survey, BC = 60 m, DC = 15 m and DE = 10 m. How wide is the river?

What should you never do when measuring shadows?

For parents and teachers

Big idea: Expectation 1.3: solving problems involving similar triangles in realistic situations (shadows, reflections, scale models, surveying).

How to help: The shadow activity is a great 15-minute outing. Make sure nobody looks at the sun. Ask your teen to sketch the two triangles and say why they are similar before calculating.

Common difficulties:

Mixing heights and shadows in the proportion (height/shadow on one side, shadow/height on the other).
Using full height instead of eye height in the mirror method.
Unit mix-ups with scales (centimetres vs metres). Convert first.
Forgetting that the sun moves: the measurements must be taken at the same time.