Lesson 4 of 5

Similarity.

About 20 minutes · 5 practice questions

Similar triangles: the same angles, every side multiplied by the same number, and the area by its square. And at the end, measuring a tree by its shadow.

Step 1 of 5 · The same shape at another size

Step 1 of 5

The same shape at another size

Two shapes are similar if one is an enlargement or a reduction of the other, like a photo and an enlargement of it. On the left is the original triangle, and a copy at another size will appear on the right. Predict what happens to the angles when every side is doubled.

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Step 2 of 5

Multiplying the sides

In similar triangles every side of the copy is longer than the matching side of the original by the same factor, $k$ ⁠, the scale factor. The copy’s vertices are named $A'$ ⁠, $B'$ ⁠ and $C'$ ⁠, so that $A'B'$ ⁠ matches $AB$ ⁠. To find $k$ ⁠, divide a side of the copy by the matching side of the original. Answer, then choose Scaled copy under Look inside to watch the copy grow.

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Step 3 of 5

Finding the scale factor

Now the other way round: the length you want in the copy is known, and you set the Scale factor. When $k$ ⁠ is more than $1$ ⁠ the copy is enlarged, and when it is less than $1$ ⁠, reduced.

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Step 4 of 5

The area grows faster

The sides grow by $k$ ⁠. What happens to the area? The original triangle’s area is printed at the bottom of the board. Predict before you look.

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Step 5 of 5

By the shadow

Similarity lets you measure tall things without climbing them. The sun is very far away, so its rays arrive almost parallel and meet the ground at the same angle. A pole and a tree standing upright, with their shadows, make two right triangles with the same angles: similar triangles. On the board, $AB$ ⁠ is the shadow and $BC$ ⁠ the height.

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Interactive scene

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Each step sets up the Scene for you, and a control marked “fixed” stays put for that step.

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Practice · Predict, then look

A copy of the triangle will appear on the right, with every side twice as long. What will its angles be?

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Practice · Calculate

In the original triangle $AB = 4$ ⁠ and $BC = 3$ ⁠. In the similar copy $A'B' = 6$ ⁠. How long is $B'C'$ ⁠?

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Practice · Reach the target

In the original triangle $AB = 4$ ⁠. Set the Scale factor so that $A'B' = 10$ ⁠ in the copy, then press Check.

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Practice · Predict, then look

The original triangle’s area is $6$ ⁠, and every side of the copy is twice as long. By what factor does the area grow?

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Practice · Calculate

A pole $2$ ⁠ metres tall casts a shadow $3$ ⁠ metres long. At the same moment a tree casts a shadow $12$ ⁠ metres long. The sun’s rays meet the ground at the same angle, so the triangles are similar. How tall is the tree, in metres?