27 . A Similar triangles have corresponding sides that are proportional to one another. In this
question, you are given that the corresponding sides in two similar triangles are in a ratio of
3:5, or . Since the length of one of the equal sides in the larger isosceles triangle is 30
centimeters, set up a proportion to find the length of one of the equal sides in the smaller
triangle. If x is the length of the side in the smaller triangle, then . Cross-multiplying
gives 5x = 90, and x = 18, as in (A). If you chose (D), you may have reversed the numbers in
the proportion and treated 30 cm as a side of the smaller triangle.
32 . J To find the area of a circle, use the area formula A = πr^2 . Given that the radius of the circle is
12 inches, you can substitute 12 into the formula to find that A = π(12)^2 = 144π. Choice (F)
multiplies the radius by π. Choice (G) is the diameter of the circle, and (H) is twice that.
Choice (K) finds the diameter of 24 inches and uses that as the radius in the area formula.
35 . A Since two sides of an isosceles triangle have equal lengths, the given side, 22 centimeters,
could be one of the equal sides, or it could be the third side of the triangle. If it were the third
side of the isosceles triangle, the lengths of the other two sides would be equal. Using the
perimeter, we know that the last two sides must total 57 − 22 = 35 cm in length. So each side
would measure = 17.5 cm, which is not in the answer choices. Therefore, you can
conclude that the side of 22 centimeters must be one of the two equal sides, and the third side
is 57 − 22 − 22 = 13 cm, as in (A). Choice (E) is rarely correct on the ACT, and in this case,
we were able to solve directly for our answer.
37 . D Given that triangle XYZ has the same base and height, you can solve for the height and base
by using the area formula for a triangle, which is A = bh. Since base and height are the
same, the equation then becomes A = h·h, or A = h^2 . Therefore, since the area of the
triangle is 72 square inches, h^2 = 72, and h = 12. Since the circle is tangent to the triangle at
point W, the line WY is both the height of the triangle and the diameter of circle O. Therefore,
the radius of the circle is 6, and you can plug this radius in to the formula for area of a circle,
which is A = πr^2 . So, A = π(6)^2 , and A = 36π. If you chose (B), be careful: this is the
circumference of the circle. If you chose (E), you may have forgotten to halve the diameter
