Question

diane wants to measure the width of a river. she marks off two right triangles, as shown in the figure. the base of the larger triangle has a length of 64 m, and the base of the smaller triangle has a length of 34 m. the height of the smaller triangle is 17.9 m. how wide is the river? round your answer to the nearest meter. (the figure is not drawn to scale.)

Explanation:

Step1: Identify Similar Triangles

The two right triangles are similar (by AA similarity, as both have a right angle and share the same angle from the diagonal). So, their corresponding sides are proportional. Let the width of the river (base of larger triangle) be \( x \). The smaller triangle has base \( 34 \) m and height \( 17.9 \) m. The larger triangle has base \( x \) and height \( 64 \) m.

Step2: Set Up Proportion

Using the property of similar triangles: \(\frac{\text{base of smaller}}{\text{height of smaller}} = \frac{\text{base of larger}}{\text{height of larger}}\), so \(\frac{34}{17.9} = \frac{x}{64}\).

Step3: Solve for \( x \)

Cross - multiply: \( 17.9x = 34\times64 \). First, calculate \( 34\times64 = 2176 \). Then, \( x=\frac{2176}{17.9}\approx121.56 \). Rounding to the nearest meter, \( x\approx122 \).

Answer:

The width of the river is approximately \(\boldsymbol{122}\) meters.