can you help me plz
Answers
Answer:
The fraction of the shaded portion is 2/3.
Step-by-step explanation:
Answer To What Fraction Is Shaded?
When you are stuck on a problem, it often helps to get started. So let’s set a goal and then work backwards. (see image 1)
The shaded fraction is equal to:
(area semicircle)/(area quarter circle)
In order to write equations for each area, let us define the following variables:
s = radius of the semicircle
q = radius of the quarter circle.
Then we have:
(area semicircle)/(area quarter circle)
= (0.5πs2)/(0.25πq2)
= 2s2/q2
So we can solve the problem if we can solve for a relationship between s2 and q2.
In such problems it often helps to connect intersection points and/or points of tangencies. So let’s get started by connecting the semicircle’s center to the points of tangency with the quarter circle. This will form a square with sides s and a diagonal s√2 (justified in the second image)
(The second diagram can be reasoned as follows. The vertical and horizontal radii of the semicircle are tangent to the quarter circle, so they form right angles. As a quarter circle also spans 90 degrees, we have a quadrilateral with 3 right angles and two sides equal to s. Thus the figure is a square, so all sides are equal to s and its diagonal is s√2.)
Now we will draw one radius of the quarter circle through the center of the semicircle. This will bisect the semicircle’s diameter–a chord of the quarter circle–so the two line segments meet at a right angle. We also draw a radius from the quarter circle to where the diameter of the semicircle meets the quarter circle. We get a figure like this: (see the third image)
Now we have a right triangle with legs s√2 and s, and a hypotenuse q, so we can use the Pythagorean Theorem to get:
(s√2)2 + s2 = q2
3s2 = q2
That’s the equation we need to solve this problem. We now go back to what we derived earlier:
(area semicircle)/(area quarter circle)
= (0.5πs2)/(0.25πq2)
= 2s2/q2
We substitute 3s2 = q2 to get:
(area semicircle)/(area quarter circle)
= 2s2/(3s2)
= 2/3
So the fraction that is shaded is 2/3. It looks like a simple geometry problem but it’s a little bit trickier than it looks!
HOPE IT HELPS U!
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