Question

check it out! example 4
a surveyor positions himself so that his line of sight to the top of a cliff and his line of sight to the bottom form a right angle as shown.
what is the height of the cliff to the nearest foot?
(image shows a cliff, a surveyor, with 28 ft and 9.5 ft labeled)

Explanation:

Step1: Identify Similar Triangles

The two right triangles (one with hypotenuse 28 ft and one with leg 9.5 ft, and the other related to the cliff height) are similar by AA (Angle - Angle) similarity (right angle and common angle). So, we can use the geometric mean theorem (altitude - on - hypotenuse theorem) which states that in a right triangle, the length of the altitude drawn to the hypotenuse is the geometric mean of the lengths of the two segments of the hypotenuse. But here, we can also set up a proportion. Let \( h \) be the height of the cliff. The two triangles are similar, so \(\frac{h}{28}=\frac{28}{9.5}\) (from the property of similar right triangles: the leg of one triangle is to the hypotenuse of the other as the hypotenuse is to the other leg).

Step2: Solve for \( h \)

From the proportion \(\frac{h}{28}=\frac{28}{9.5}\), we can cross - multiply to get \( h=\frac{28\times28}{9.5}\).
First, calculate \( 28\times28 = 784 \). Then, divide by 9.5: \( h=\frac{784}{9.5}\approx82.526\). Rounding to the nearest foot, we get \( h\approx83 \).

Answer:

The height of the cliff to the nearest foot is \(\boldsymbol{83}\) feet.