Question

The figure below shows a triangular lawn labeled PQR. There are two fences, AB and BC. What is the total length of the fence AB and BC?

Answer 1

128 feet

56 feet

100 feet

72 feet

Answer 2

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Answer 3

I am going to tell you one way to work this problem, but there may be several other ways, some may be simpler than the way I did it.

Use the Law of Cosines and solve for angle Q. If you are not familiar with the Law of Cosines it is: \[a ^{2}=b ^{2}+c ^{2}-2bcCos A\] Since all three sides of triangle PQR are given, it is possible to obtain angles, Q, P, and R. Angle R is the same as angle C and A.
For angle Q: 120^2 = 100^2+140^2-2(100)(140)Cos A
Cos Q=(14400-10000-19600)/28000
Cos Q =0.54285 Inverse Cos Q= 57.122 degrees Angle Q is 57.122 degrees.

In a similar manner solve for Angle P, you should get arounds 78.463 degrees
Do the same for angle R, which is also angle C, getting approx 44.44 degrees.

Now calculate the BC fence length using the Law of Sines; the Law of sines is:
a/Sin A = b/Sin B =c/Sin C
60/Sin 44.44 = BC/Sin 57.122 degres
BC= 72 feet

I will leave AB up to you but using those two laws and solving for various parts you can do it. Note: that PB is 40 ft. Also AB can be solved by solving for CQ and subtract that from 140 ft.

Answer 4

Calculate QC. We now have BC, angle C, and angle Q. Angle QBC is equal to
180-44.4-57.122= 78.44 degrees.
Now to the Law of Sines: (CQ/Sin 78.46) = (72)/(Sin 57.122)= 84 Ft.
Now you can calculate RC which is 140-84 = 56 ft.
by the laws of parallegram then AB = 56 ft.