Question

A surveyor, Toby, measures the distance between two landmarks and the point where he stands. He also measured the angles between the landmarks in degrees.
What is the distance, x, between the two landmarks? Round the answer to the nearest tenth.

Answer 1

What we know so far:
Side 1 = 55m
Side 2 = 65m
Angle 1 = 40°
Angle 2 = 30°

What we are looking for:
Toby's Angle = ?
The distance x = ?

We need to look for Toby's angle so that we can solve for the distance x by assuming that the whole figure is a SAS (Side Angle Side) triangle.

Solving for Toby's Angle:
We know for a fact that the sum of all the angles of a triangle is 180°; therefore,
180° - (Side 1 + Side 2) = Toby's Angle
Toby's Angle  = 180° - (40° + 30°)
Toby's Angle = 110°

Since we already have Toby's angle, we can now solve for the distance x by using the law of cosines r² = p²+ q²− 2pq cos R where r is x, p is Side1, q is Side2, and R is Toby's Angle.

x² = Side1² + Side2² - 2[(Side1)(Side2)] cos(Toby's Angle)
x² = 55² + 65² - 2[(55)(65)] cos(110°)
x² = 3025 + 4225 -7150[cos(110°)]
x² = 7250 - 2445.44
x = √4804.56
x = 69.31m

∴The distance, x, between two landmarks is 69.31m