Question

A tunnel must be made through a hill. As a result, a surveyor and an engineer create a sketch of the area. The sketch, displayed below, includes information they have either researched or measured. They need to build a tunnel from the point E to the point H on the sketch. Calculate the distance from E to H. When similar triangles are used, explain how you know they represent similar triangles before performing the calculation.

Answer 1

498 m

Explanation:

The AAA theorem states that triangles are similar if all three corresponding angles are equal.

1. Compare triangles FHS and ILS

(a) Reason for similarity

∠F = ∠I = 90°

∠S is common.

∴ ∠H = ∠L

(b) Calculate SL

`\begin{array}{rcl}(SF)/(SH) & = & (SI)/(SL)\\\\(225)/(380) & = & (225 + 475)/(SL)\\\\225SL & = & 380 * 700\\& = & 266000\\SL & = & \textbf{1182 m}\\\end{array}`

2. Compare triangles ILS and GLE

(a) Reason for similarity

∠I = ∠G = 90°

∠L is common.

∴ ∠S = ∠E

(b) Calculate LE

`\begin{array}{rcl}(IS)/(GE) & = & (LS)/(LE)\\\\(700)/(180) & = & (1182)/(LE)\\\\700LE & = & 180 * 1182\\& = & 212800\\LE & = & \textbf{304.0 m}\\\end{array}`

3. Calculate EH

LE + EH + HS = LS

304.0 m + EH + 380 m = 1182 m

EH + 684 m = 1182 m

EH = 498 m

The distance from E to H is 498 m.