I'm curious on the two questions below - why is is that in the first one we can assume 30-60-90 but in the second one when we are told the diagonal is 13, we can’t assume 30-60-90 in order to answer? (I had answered D for the first one, and D for the second one)
1. Circle O is inscribed in equilateral triangle ABC, which is itself inscribed in circle P. What is the area of circle P?
(1) The area of circle O is 4π.
(2) The area of triangle ABC is 12√2
We're not truly assuming that it's 30-60-90. Since it's an equilateral triangle, we know that drawing a height is guaranteed to give us a 30-60-90 triangle.
1) With the area of O, we can find a radius of 2. With our knowledge of equilateral triangles, we can create a 30-60-90 triangle where the radius of 2 is the side opposite the 30-degree angle. The hypotenuse then becomes 4; conveniently for us, the hypotenuse that we drew is also the radius of circle P, so we can find the area. Statement 1 is sufficient on its own, so we can eliminate B, C, and E.
2) I'm going to guess that you meant the area of ABC is 12√3? If so, we can use the equilateral area formula to find that each side of the triangle is 4√3; from there, we can derive the same 30-60-90 triangle as in Statement 1, and once we find the hypotenuse of the triangle, the area of Circle R is pretty easy to find. Statement 2 is also sufficient, so our answer is D.
Hopefully my diagram makes sense; I drew it with my finger on my iPad.
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