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A circumscribed triangle with a side as a diameter has a 90 degree angle.
Show Answer Explanation
- The question asks us to determine the area of the shaded region, while giving us the circumference of the circle in which the triangle is inscribed. The way to answer this problem is to determine the area of the entire circle, and subtract the area of half the circle (the bottom half without the triangle in it) and subtract from this the area of the triangle. What we have left is the area of the shaded region.
- The circumference of a circle is *diameter. So the diameter, then, must be 8.
Circumference = *diameter
8 = D
D = 8
- The radius is half of the diameter, which comes to 4. The area of a circle is equal to r 2. Therefore the area is 4 2. or 16 .
- Any diameter divides a circle into two equal halves, so the bottom half of the circle has an area of 8
- To determine the area of the triangle, first note that the area of a triangle is 1/2 its base multiplied by its height. The base here is the same as the diameter of the circle, because line AC goes through the center of the circle, O. We have determined earlier on that the diameter is 8, so the base is also 8.
- To ascertain the height of the triangle, notice that triangle ABC is an isosceles triangle (with two sides of the triangle of equal length). This means that the height of this triangle is the length of the line from the vertex B to the midpoint of the base, line AC. The midpoint of the base also happens to be the center of the circle, O, meaning that the line OB is not only the height of the circle, but also the radius. Therefore the height is 4, or half of the diameter, which is 8.
- Taking our formula for area of a triangle we have 1/2(8)(4) = 16. So the area of the triangle is 16.
- Now we have the area of the circle, the area of the triangle, and the area of the bottom half of the circle. Finally, we subtract the sum of 1). the area of the bottom half of the circle and 2). the area of the triangle from the area of the entire circle. Thus the area of the shaded region is:
16 – (8 + 16) 16 – 8 – 16
8 – 16, answer choice (E).