Lesson 9-5: Indirect Measurement

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.

Property

Similar triangles can be used to find unknown measurements indirectly. When objects and their shadows are measured at the same time, they form similar right triangles because the sun's rays are parallel. By setting up a proportion using corresponding sides of these similar triangles, we can solve for unknown heights or distances.

Examples

Property

When geometric figures are similar, the ratios of their corresponding sides are equal. This allows us to set up a proportion to find an unknown side length, such as $\frac{\text{Height}_A}{\text{Shadow}_A} = \frac{\text{Height}_B}{\text{Shadow}_B}$.

Examples

• A vertical 3-meter pole casts a 5-meter shadow. A nearby building casts a 35-meter shadow. The building's height (h) is found with $\frac{h}{3} = \frac{35}{5}$, so $h=21$ meters.
• A model airplane is 1 ft long and its real-life counterpart is 40 ft long. If the model's wingspan is 0.8 ft, the real plane's wingspan (w) is $\frac{w}{0.8} = \frac{40}{1}$, so $w=32$ ft.

Explanation

Think of proportions as a balancing act for shapes. If you know three out of four related measurements from two similar figures, you can always find the missing fourth piece of the puzzle. Just match up the corresponding sides to create equal ratios, cross-multiply, and solve for the mystery length!