A semicircle is constructed along each side of a right triangle with legs 6 inches and 8 inches. The semicircle placed along the hypotenuse is shaded, as shown. What is the total area of the two non-shaded crescent-shaped regions? Express your answer in simplest form.
The real answer is 24
A+B=C, the Pythagorean theorem states, in this case,
we want to minus C from the area of the two shaded areas that are outside but this will just end up in the right triangles area, so,
it will be 6*8/2=24
area of large semi-circle = 25π/2
haha very funny you dont even know what their account name is.
That is wrong. That is not the answer.
lmaooooo
To find the total area of the two non-shaded crescent-shaped regions, we need to subtract the area of the shaded semicircle from the combined areas of the two semicircles.
First, let's find the area of each semicircle. The formula to calculate the area of a semicircle is:
Area = (π * r^2) / 2
where π is a constant (approximately equal to 3.14159) and r is the radius.
For the semicircles along the legs of the right triangle, the radius is half the length of the leg. In this case, the radius is 6/2 = 3 inches.
So, the area of each semicircle is:
Area = (π * 3^2) / 2 = (π * 9) / 2
Next, let's find the area of the shaded semicircle along the hypotenuse. To do this, we need to find the length of the hypotenuse. We can use the Pythagorean theorem to find the length of the hypotenuse, which states:
c^2 = a^2 + b^2
where c is the length of the hypotenuse, and a and b are the lengths of the legs of the right triangle.
In this case, a = 6 inches and b = 8 inches. So we have:
c^2 = 6^2 + 8^2
c^2 = 36 + 64
c^2 = 100
c = √100
c = 10 inches
The radius of the shaded semicircle is half the length of the hypotenuse, which is 10/2 = 5 inches.
So, the area of the shaded semicircle is:
Area = (π * 5^2) / 2 = (π * 25) / 2
Now, we can find the combined areas of the two non-shaded crescent-shaped regions:
Combined area = 2 * (Area of each semicircle) - Area of shaded semicircle
Combined area = 2 * ((π * 9) / 2) - ((π * 25) / 2)
Simplifying this expression, we get:
Combined area = (π * 9) - (π * 25)
Combined area = π * (9 - 25)
Combined area = π * (-16)
Therefore, the total area of the two non-shaded crescent-shaped regions is -16π square inches.
Argghh, of course, I think I need a nap.
length of hypotenus = 10
(6^2 + 8^2 = 10^2 )
So the radius of the large semicircle = 5
area of large semi-circle = 25π
so the total area of the two non-shaded regions is also 25π in^2