Indirect Measurement Math Example 4

Follow the full solution, then compare it with the other examples linked below.

medium

To measure river width, a surveyor marks

A

and

B

on one bank with

AB = 80

m. A tree

T

on the other bank is directly opposite

A

(so

\angle TAB = 90°

). From

B

, the angle

\angle TBA = 52°

. Find width

AT

1
In right triangle $ABT$ with the right angle at $A$ : $AB = 80$ m is adjacent to $\angle B = 52°$ , and $AT$ is opposite to $\angle B$ .
2
Use $\tan 52° = \dfrac{AT}{AB} = \dfrac{AT}{80}$ .
3
Solve: $AT = 80 \times \tan 52° \approx 80 \times 1.2799 \approx 102.4$ m.

Answer

The river width is approximately

102.4

Setting up a right triangle with a known side and angle lets trigonometry compute the inaccessible distance. The right angle at

A

(standing directly opposite

T

) simplifies the configuration to a single tangent equation.

About Indirect Measurement

Indirect measurement finds unknown lengths by using proportional relationships instead of direct measuring tools.

A tree casts a shadow of [formula] m at the same time a [formula] m pole casts a shadow of [formula]

From a point [formula] m from the base of a building, the angle of elevation to the top is [formula]

A [formula]-foot person casts a [formula]-foot shadow. At the same time, a nearby flagpole casts a [