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janiyahmcgolley
23.06.2020 •
Mathematics
What is the area of the shaded region in the figure below ? Leave answer in terms of pi and in simplest radical form
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Ответ:
Step-by-step explanation:
That shaded area is called a segment. To find the area of a segment within a circle, you first have to find the area of the pizza-shaped portion (called the sector), then subtract from it the area of the triangle (the sector without the shaded area forms a triangle as you can see). This difference will be the area of the segment.
The formula for the area of a sector of a circle is:
Filling in:
Now for tricky part...the area of the triangle.
We see that the central angle is 60 degrees. We also know, by the definition of a radius of a circle, that 2 of the sides of the triangle (formed by 2 radii of the circle) measure √3. If we pull that triangle out and set it to the side to work on it with the central angle at the top, we have an equilateral triangle. This is because of the Isosceles Triangle Theorem that says that if 2 sides of a triangle are congruent then the angles opposite those sides are also congruent. If the vertex angle (the angle at the top) is 60, then by the Triangle Angle-Sum theorem,
180 - 60 = 120, AND since the 2 other angles in the triangle are congruent by the Isosceles Triangle Theorem, they have to split that 120 evenly in order to be congruent. 120 / 2 = 60. This is a 60-60-60 triangle.
If we take that extracted equilateral triangle and split it straight down the middle from the vertex angle, we get a right triangle with the vertex angle half of what it was. It was 60, now it's 30. The base angles are now 90 and 60. The hypotenuse of this right triangle is the same as the radius of the circle, and the base of this right triangle is
. Remember that when we split that 60-60-60 triangle down the center we split the vertex angle in half but we also split the base in half.
Using Pythagorean's Theorem we can find the height of the triangle to fill in the area formula for a triangle which is
Finding the height value using Pythagorean's Theorem:
Taking the square root of both the 9 and the 4 (which are both perfect squares, thankfully!), we get that the height is 3/2. Now we can finally fill in the area formula for the triangle!
Therefore, the area in terms of pi for that little segment is
Ответ:
the answer is c
Step-by-step explanation:
because it scales up and moves to the right