Question

The distance from Eric to Gavin is 20 feet, and the distance from Eric to Quinn is 22 feet. A triangle is drawn connecting the positions of Eric, Quinn, and Gavin. How far is Gavin from Quinn if the angle marking Eric’s position is 26°?
40.9 ft

9.6 ft

16.7 ft

22.1 ft

Answer 1

9.6 ft is the correct answer because sin(26)= x/22

Answer 2

Refer to the attached image.

Assume that Eric is standing at position 'A', Gavin is standing at position 'B' and Quinn is standing at the position 'C'.

Since, the distance from Eric to Gavin(AB) is 20 feet, and the distance from Eric to Quinn(AC) is 22 feet.

The angle marking Eric’s position($\angle A$) is 26°

By law of cosines, which states:

"In triangle ABC, with angles A, B and C. Side opposite to angle A is 'a', side opposite to angle B is 'b' and side opposite to angle C is 'c'. This formula is used to find the third side of a triangle when we know two sides and the angle between them. Therefore, it states:

$a^{2}=b^{2}+c^{2}-2bc\cos A$

In triangle ABC, c=20 feet , b=22 feet and angle A = 26 degrees.

$a^{2}=(22)^{2}+(20)^{2}-(2 \times 22 \times 20 \times \cos 26^{\circ})$

$a^{2}=93.06$

$a=\sqrt{93.06}$

a=9.64 feet

a=9.6 feet

The distance from Gavin to Quinn is 9.6 feet.

Therefore, Option B is the correct answer.