Question

use similar triangles and proportions to find the length of the swamp shown in the figure below. (assume that ( ell_1 = 100 mathrm{ft}, ell_2 = 120 mathrm{ft} ), and ( h_1 = 10 mathrm{ft} ).) ft resources read it watch it

Explanation:

Step1: Set up the proportion using similar triangles

Since the two triangles are similar, the ratios of their corresponding sides are equal. Let \(h_2\) be the height of the larger triangle (length of the swamp). We have the proportion \(\frac{h_1}{\ell_1}=\frac{h_2}{\ell_1 + \ell_2}\).

Step2: Substitute the given values

Substitute \(\ell_1 = 100\) ft, \(\ell_2=120\) ft, and \(h_1 = 10\) ft into the proportion: \(\frac{10}{100}=\frac{h_2}{100 + 120}\).

Step3: Solve for \(h_2\)

Cross - multiply: \(100h_2=10\times(100 + 120)\).
First, calculate \(10\times(100 + 120)=10\times220 = 2200\).
Then, \(h_2=\frac{2200}{100}=22\).

Answer:

\(22\)