Lesson 65: Applications Using Similar Triangles

New Concept

We can find the measure of objects that are difficult to measure by using similar triangles. The side lengths of similar triangles are proportional.

What’s next

Next, you’ll tackle real-world problems, calculating heights and distances by setting up and solving proportions for similar triangles.

Property

In geometry, "similar" (denoted by $\sim$) has a very strict mathematical meaning. Two figures are similar if they have the exact same shape, but not necessarily the same size. For this to be true, two rules must be met simultaneously:

1. All corresponding angles must be exactly congruent (equal).
2. All corresponding side lengths must be proportional (they share the same scale factor, $k$).

Examples

• The Blueprint: A floor plan and the actual house are similar. If a room's corner is $90^\circ$ on the paper, it must be exactly $90^\circ$ in the real house.
• Finding Missing Sides: $\Delta ABC \sim \Delta XYZ$. If side $AB = 4$ and $BC = 6$, and the corresponding side $XY = 8$, we know the scale factor is $2$ (because $8 / 4 = 2$). Therefore, side $YZ$ must be $6 \times 2 = 12$.

Explanation

Think of similar shapes as perfect zooming in or out. The angles act as the "skeleton" that keeps the shape from distorting, which is why angles NEVER change during a zoom. The sides act as the "muscles" that stretch or shrink proportionally. If one side doubles in length, every other side must also double, otherwise the shape warps and is no longer similar!

Property

Indirect measure is a clever technique that uses properties of similar triangles, like proportional side lengths, to calculate the size of an object that is difficult or impossible to measure directly.

Examples

• To find a flagpole's height (h), you measure its shadow (24 m) and the shadow of a 2 m stick (1.6 m). The proportion is $\frac{h}{24} = \frac{2}{1.6}$, so $h=30$ m.
• You find the distance (D) across a canyon by creating a smaller similar triangle. With a scale factor of 50 and a measured small side of 10 yards, $D = 10 \times 50 = 500$ yards.

Explanation

Why would you climb a scary-tall tree or try to swim across a wide river with a tape measure? Indirect measurement is your mathematical superpower! It lets you find huge heights and distances safely from the ground by measuring smaller, more convenient things like shadows or stakes.