Earlier this week, we posted a challenging ACT Math geometry problem. Here it is again if you missed it (then head down below to find the answer and explanation):
ACT Challenge Question #3
As shown in the figure below, A is the center of the circle, and right triangle ABC intersects the circle at D and E. Point D is the midpoint of AC, which is 22 cm long. The shaded region inside the circle and outside the triangle has an area of square centimeters. What is the measure of angle B?
Answer and Explanation
ANSWER: E, 57°
The ACT rarely gives you any unnecessary information in a math word problem. This means that all of the details in the question give you important clues that you need to solve the problem. Here’s another important tip: Whenever you are dealing with circles, and you aren’t given the radius, your first step should be to find the radius. The radius is the key to unlocking other important circle things, such as the area, circumference, sector area, or arc length.
In the case of this question, we know that AC is 22 cm, that D is the midpoint of AC, and that D serves as a point of intersection between the circle and the triangle. This means that AD is the radius and should be half of 22 cm, or 11 cm. Knowing that 11 cm is the radius allows us to find the area of the entire circle using the equation .
We are told in the problem that the area of the shaded region is . This means that the area of the unshaded sector of the circle inside the triangle must be , since these two regions must add up to the total area of the circle: .
This information helps us find the fraction of the circle delimited by the triangle. Because a sector is a fraction of a circle, we can use the proportion of the area of the sector to the area of entire circle to find the degree measure of the central angle.
Because every circle has 360 degrees:
Solving this proportion to find angle A gives us x = 32.727272 repeating, or approximately 33°.
Since we know one of the other angles of the triangle is 90°, we can find the measure of the remaining angle, angle B, by subtracting the two known angles from 180° (since the angles in a triangle always add up to 180°).
180 – 90 – 33 = 57
So our answer is (E) 57°.
And for some bonus fun (particularly if you got tripped up on finding the area of the sector), check out this cool animated video we made on this topic!