Question

question
to indirectly measure the distance across a lake, jose makes use of a couple landmarks at points g and h. he measures fi, ig, and ij as marked. find the distance across the lake (gh), rounding your answer to the nearest hundredth of a meter.
(diagram is not to scale.)

Explanation:

Step1: Identify Similar Triangles

Triangles \( \triangle F IJ \) and \( \triangle FGH \) are similar (by AA similarity, as both have a right angle and share \( \angle F \)). So, the ratios of corresponding sides are equal: \( \frac{IJ}{GH} = \frac{FI}{FG} \).

Step2: Substitute Known Values

We know \( IJ = 92.8 \, \text{m} \), \( FI = 155 \, \text{m} \), \( FG = FI + IG = 155 + 110 = 265 \, \text{m} \). Let \( GH = x \). Then \( \frac{92.8}{x} = \frac{155}{265} \).

Step3: Solve for \( x \)

Cross - multiply: \( 155x = 92.8\times265 \). Calculate \( 92.8\times265 = 92.8\times(200 + 60+5)=92.8\times200+92.8\times60 + 92.8\times5=18560+5568 + 464 = 24592 \). Then \( x=\frac{24592}{155}\approx158.66 \).

Answer:

\( 158.66 \) meters