In triangle PQR, we have <P = 30 degrees, <Q = 60 degrees, and <R=90 degrees. Point X is on line PR such that line QX bisects <PQR. If PQ = 12, then what is the area of triangle PQX?

Question

In triangle PQR, we have <P = 30 degrees, <Q = 60 degrees, and <R=90 degrees. Point X is on line PR such that line QX bisects <PQR. If PQ = 12, then what is the area of triangle PQX?

I have determined that i need to use the equation PR/\sqrt{3}, but i cannot find PR.

Answer 1

So you have the famous 30-60-90 right-angled triangle, which has sides in the ratio

1 : √3 : 2
(memorize those values, easy to remember that the smallest side is opposite the smallest angle and the largest side is opposite the largest angle)

so using simple ratios
QR : PR : 12 = 1 : √ 3 : 2

QR/1 = 12/2 ---> QR = 6
PR/√3 = 12/2 ---> PR = 12√3/2 = 6√3
(did you notice that we simply multiplied each of the ratio terms by 6, keeping our new triangle similar to 1:√3:2 )

area of PQR = (1/2)PR*QR
= (1/2)(6√3)(6) = 18√3

Since PXR has the same height, but only half the base ,
its area is 9√3

Answer 2

x is not the midpoint of PR, so theres a bit more math to do. I don't believe 9 sqrt(3) is correct.

Answer 3

To find the length of PR, we can use the Law of Sines. In triangle PQR, we have angles <P = 30 degrees, <Q = 60 degrees, and <R = 90 degrees. Let's denote the length of PR as x.

According to the Law of Sines:
PR/sinQ = QR/sinP

Substituting the known values:
x/sin(60) = QR/sin(30)

Now, using the sine values of 60 degrees and 30 degrees:
x/(√3/2) = QR/(1/2)
2x/√3 = QR

Since triangle PQR is a 30-60-90 right triangle, we know that QR equals 12 (since PQ = 12). Thus, we can solve for x:
2x/√3 = 12
2x = 12√3
x = 6√3

Now, we have found the length of PR, which is equal to 6√3.

Next, we want to find the area of triangle PQX. We know that triangle PQX is a right triangle, where PQ is the base and QX is the height.

Given that PQ = 12 and QX bisects <PQR, we can see that triangle PQX is a 30-60-90 triangle. In this triangle, the ratio of the sides is 1:√3:2.

Therefore, the ratio of PQ to QX is also 1:√3. Since PQ = 12, QX can be found by dividing PQ by √3:
QX = PQ/√3 = 12/√3 = 4√3

Now, we can calculate the area of triangle PQX using the formula for the area of a triangle:
Area = (base × height)/2

Plugging in the values:
Area of triangle PQX = (12 × 4√3)/2 = 24√3

So, the area of triangle PQX is 24√3.

In triangle PQR, we have <P = 30 degrees, <Q = 60 degrees, and <R=90 degrees. Point X is on line PR such that line QX bisects <PQR. If PQ = 12, then what is the area of triangle PQX?

Gggyy

So you have the famous 30-60-90 right-angled triangle, which has sides in the ratio

x is not the midpoint of PR, so theres a bit more math to do. I don't believe 9 sqrt(3) is correct.

To find the length of PR, we can use the Law of Sines. In triangle PQR, we have angles <P = 30 degrees, <Q = 60 degrees, and <R = 90 degrees. Let's denote the length of PR as x.