Given: ∆PQR, m∠R = 90°
m∠PQR = 75°
M ∈ PR , MP = 18
m∠MQR = 60°
Find: RQ
M is on segment PR
9sqrt3 is wrong(the system did not take it)
is not 9sqrt 3 is just 9
To find the length of RQ, we can use the cosine rule. The cosine rule states that in a triangle, the square of one side is equal to the sum of the squares of the other two sides minus twice their product multiplied by the cosine of the included angle.
In this case, we have a right triangle ∆PQR with a right angle at R. Let's label the length of RQ as x.
Using the cosine rule, we can write the equation:
x^2 = PQ^2 + QR^2 - 2(PQ)(QR)cos(∠PQR)
From the given information, we know that ∠PQR = 75° and PQ is the side opposite to it. Therefore, we need to find the lengths of PQ and QR.
To find PQ, we can use the sine rule, which states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all three sides.
Using the sine rule with ∆PQR, we have:
PQ/sin(75°) = QR/sin(90°)
Since sin(90°) = 1, we can simplify the equation to:
PQ = QR*sin(75°)
To find QR, we can use the cosine rule again. From the given information, we know that ∠MQR = 60° and MQ = 18.
Using the cosine rule, we have:
QR^2 = MQ^2 + MR^2 - 2(MQ)(MR)cos(∠MQR)
Since the length of MR is not given directly, we need to find it using the Pythagorean theorem in ∆MPR, which is a right triangle with ∠M = 90°:
MP^2 + MR^2 = PR^2
18^2 + MR^2 = PQ^2
324 + MR^2 = PQ^2
We already found that PQ = QR*sin(75°), so the equation becomes:
324 + MR^2 = (QR*sin(75°))^2
Now we have two equations:
PQ = QR*sin(75°)
324 + MR^2 = (QR*sin(75°))^2
We can substitute the value of PQ in the second equation:
324 + MR^2 = (QR*sin(75°))^2
324 + MR^2 = (QR^2*sin^2(75°))
324 + MR^2 = QR^2*(sin^2(75°))
To solve for QR, we need one more equation involving QR. We can use the fact that M is on segment PR.
Since MR + MQ = PQ, we have:
MR + 18 = PQ
MR = PQ - 18
Substituting this into the equation, we get:
324 + (PQ - 18)^2 = QR^2*(sin^2(75°))
Now we have three equations:
PQ = QR*sin(75°)
324 + (PQ - 18)^2 = QR^2*(sin^2(75°))
MR + 18 = PQ
Solving these equations simultaneously will give us the value of QR. Once we find QR, we can calculate RQ by subtracting it from PQ.