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. a) Prove the three triangles ΔSFH, ΔEGL, and ΔSIL are similar. b) Use similar triangles to determine the length from H to E.

Answer 1

a) Look step by step explanation

b) x = 498,22

Explanation:

a) Two triangles are similar if all their angles are equal

Then in Δ SFH and Δ GEL we got:

∡ SFH = ∡ EGL right angles

∡ LEG = ∡ HSF since the same straight line (SL) is cut by parallel lines (vertical segments EG and SF )

∡ SHF = ∡ ELG The same straight line SL is cut by two parallel segments FH and GL ( horizontal segments)

The Δ SIL is similar to the two previous Δs according to:

∡ SIL is a right angle

∡ ILS = ∡ GLE = ∡ FHS

∡ ISL = ∡FSH = ∡GEL

The condition for similar triangles is satisfied

b)

From Δs FSH and GEL

SF/EG = 380/EL

225/180 = 380/EL

225*EL = 180*380

EL = 180*380/225

EL = 304

From Δs SIL and SFH

225/ (225 +475) = 380/ (380 + x + 304)

225 / 700 = 380 / 684 + x

225 * ( 684 + x ) = 700*380

153900 + 225*x = 266000

225*x = 266000 - 153900

225*x = 112100

x = 498,22