Integrated Math 2 / Unit 6: Similarity
6.5 Indirect Measurement and Right Triangles
Notes
Similar triangles let you find heights and distances you cannot measure directly.
- Sunlight makes similar triangles: $\displaystyle \frac{\text{object height}}{\text{object shadow}} = \frac{\text{person height}}{\text{person shadow}}$.
- Example: a 6 ft person has a 4 ft shadow. A tree has a 20 ft shadow. $\displaystyle \frac{6}{4} = \frac{h}{20}$ gives $h = 30$ ft.
- In a right triangle, the altitude to the hypotenuse makes two triangles similar to the whole triangle. Altitude $\displaystyle = \sqrt{(\text{segment 1})(\text{segment 2})}$.
Practice
A 6 ft person casts a 4 ft shadow. A tree casts a 20 ft shadow at the same time. How tall is the tree?
$\displaystyle \frac{6}{4} = \frac{h}{20}$ gives $4h = 120$, so $h = 30$.
A 2 m stick casts a 3 m shadow. A pole casts a 12 m shadow at the same time. How tall is the pole?
$\displaystyle \frac{2}{3} = \frac{h}{12}$ gives $3h = 24$, so $h = 8$.
In a right triangle, the altitude to the hypotenuse splits it into segments of length 4 and 9. How long is the altitude?
The altitude is the geometric mean: $\displaystyle \sqrt{4 \cdot 9} = 6$.
Ready for more? Take the Unit 6 test.