Similarity and scale factor (Geometry) · lesson

One scale factor for every length

💡 Find the scale factor from one pair of matching sides, then use it on every other length.

In similar figures, every length is multiplied by the same number. That number is the scale factor. To find it, divide a length on the new figure by the matching length on the old one.

Perimeter is a length too, so it grows by the same scale factor.

A map scale works the same way. If 1 cm stands for 50 km, multiply map centimeters by 50 to get real kilometers.

🧠 Let's work it out

A 3 cm by 5 cm rectangle is similar to a bigger rectangle whose short side is 12 cm. Find the long side.

Match short sides to get the scale factor: 12 ÷ 3 = 4.
Use it on the long side: 5 × 4 = 20.

What do you think comes next?

similar: same shape, possibly a different size
scale factor: the number every length is multiplied by
perimeter: the distance around a figure

🔗 Khan Academy: Similarity · Khan Academy (free) ↗

Spotting similar triangles

💡 When angles match, the triangles are similar, so matching sides form equal ratios.

Sun rays hit a person and a tree at the same angle. So each object, its shadow, and the sun ray form similar triangles. Height ÷ shadow is the same for both.

A line drawn parallel to one side of a triangle cuts off a smaller, similar triangle. It also splits the other two sides in the same ratio: top piece ÷ bottom piece is equal on both sides.

In a right triangle, the height drawn to the long side splits that side into two pieces. If the pieces are p and q, the height h fits p/h = h/q, so h² = p × q.

🧠 Let's work it out

A 5-foot post casts a 2-foot shadow. A building nearby casts a 14-foot shadow at the same time. How tall is the building?

Set height over shadow equal: 5/2 = h/14.
Multiply both sides by 14: h = 5 × 14 ÷ 2 = 35.
The building is 35 feet tall.

What do you think comes next?

parallel: lines that never meet, always the same distance apart
ratio: a comparison made by dividing one amount by another
altitude: a height drawn straight down to a side, making a right angle

Areas grow by the square

💡 If lengths are multiplied by k, areas are multiplied by k².

Area has two dimensions, length and width. Both grow by the scale factor, so area grows by the scale factor times itself.

Going the other way, if you know the area ratio, take its square root to get the scale factor for lengths.

🧠 Let's work it out

Two similar triangles have areas 9 and 49. A side of the small one is 12. Find the matching side of the big one.

Area ratio = 49 ÷ 9.
The scale factor is the square root: √49 ÷ √9 = 7/3.
The matching side is 12 × 7/3 = 28.

What do you think comes next?

square root: the number that times itself gives the value: √49 = 7

Dilations from a center

💡 A dilation scales each point's distance from the center by the scale factor.

A dilation centered at the origin multiplies both coordinates by the scale factor. A factor of 1/3 shrinks the figure. A factor of 2 doubles it.

If the center is another point, find the step from the center to the point. Scale that step. Then add the center back.

🧠 Let's work it out

Dilate (8, 4) by a factor of 1/3, centered at (2, 1).

The step from the center is (8 − 2, 4 − 1) = (6, 3).
Scale it: (6 × 1/3, 3 × 1/3) = (2, 1).
Add the center back: (2 + 2, 1 + 1) = (4, 2).

What do you think comes next?

dilation: resizing a figure from a center point by a scale factor
center of dilation: the fixed point the figure grows or shrinks from