Given: ∆PQR, m∠R = 90°
m∠PQR = 75°
M ∈
PR
, MP = 18
m∠MQR = 60°
Find: RQ
from what you know about 30-60-90 right triangles, if we let x = RQ then
MR = x√3
Now, we have
(18+x√3)/x = tan 75°
So solve for x, which is RQ.
It might help if you recall that tan75° = 2+√3
To find the length of RQ in triangle PQR, we can use the Law of Cosines. The Law of Cosines states that in any triangle with sides of lengths a, b, and c, and angle C opposite side c, the following equation holds:
c^2 = a^2 + b^2 - 2ab * cos(C)
In this case, we can label the sides of triangle PQR as follows:
PR = a
QR = b
PQ = c
Based on the given information, we know:
PR = MP = 18
m∠R = 90°
m∠PQR = 75°
m∠MQR = 60°
We are trying to solve for QR, which is equivalent to c in the Law of Cosines equation.
First, let's find the measure of the angle ∠RPQ. Since the sum of the angles in a triangle is 180°, we can calculate:
m∠RPQ = 180° - m∠PQR - m∠R
= 180° - 75° - 90°
= 15°
Next, we can use this angle and the known side lengths to apply the Law of Cosines:
QR^2 = PQ^2 + PR^2 - 2 * PQ * PR * cos(RPQ)
Substituting the values we know:
QR^2 = (18)^2 + (18)^2 - 2 * 18 * 18 * cos(15°)
Now, we can calculate QR:
QR^2 = 324 + 324 - 648 * cos(15°)
QR^2 = 648 - 648 * cos(15°)
QR^2 = 648 * (1 - cos(15°))
QR = √(648 * (1 - cos(15°)))
Calculating this value will give us the length of RQ.