In , U is the midpoint of , V is the midpoint of , and W is the midpoint of . Angle R is the bottom left, angle T is the bottom right, and angle S is the top. Inside is another equilateral triangle facing down. Angle U is on the left, angle V is on the right, and angle W is the bottom. This forms four triangles within the larger triangle. Triangle URW has side lengths 12 and 15.9. Triangle VWT has side lengths 3y and 11. Triangle SUV has side lengths blank and 2x. Question Multiple Choice Use the Triangle diagram to answer the question. What is the length of ? (1 point)
The length of UV is equal to the length of TS because they are both midpoints of the same side of the larger triangle. Since angle S is an equilateral triangle, we know that the angles in that triangle are all equal to 60 degrees.
We can use the Law of Cosines to find the length of ST. Let's call it "a". Using the side lengths of triangle URW, we have:
a^2 = 12^2 + 15.9^2 - 2(12)(15.9)cos(60)
a^2 = 144 + 252.81 - 2(12)(15.9)(1/2)
a^2 = 144 + 252.81 - 12(15.9)
a^2 = 144 + 252.81 - 190.8
a^2 = 206.01
So, the length of ST is sqrt(206.01).
Since UV is parallel to ST, we have congruent triangles VST and UST. Also, since UV = ST, we know that VU = TS = sqrt(206.01).
Therefore, the length of VU is sqrt(206.01).
Answer: sqrt(206.01)