Answer :
Answer:
Area=36[tex]\pi[/tex]-72 [tex]cm^{2}[/tex] Perimeter=12[tex]\sqrt{2}[/tex] + 6[tex]\pi[/tex] cm
Step-by-step explanation:
To get the area of the shaded region, we must first calculate the whole area, and then find the area of the triangle. After that, we have to subtract the area of the triangle from the whole area to get the area of the shaded region. Let's begin now.
Area of the whole:
This shape is 1/4th of a circle, which can be seen by the 90-degree angle in the top right of the image. Since 90 is 1/4th of 360 that means the area of this shape is 1/4th the area of a circle with radius 12. We can see that the radius is 12 as AB is 12, and B is the center.
That means the area of the whole is:
1/4*[tex]\pi[/tex]*r*r, where r is the radius. Since we know the radius is 12, we can plug it in.
1/4*[tex]\pi[/tex]*144, we can simplify this by doing 144/4, which gives us
36[tex]\pi[/tex] as the area of the whole.
Now, the area of the triangle is 1/2*b*h, where b is the base and h is the height. Since in a right triangle the legs are the base and height, you can set up the equation below.
1/2*12*12
6*12
72
After some simplification, we get 72 as the area of the triangle. Now all we have to do is subtract.
Area of the whole - Area of the triangle = Area of the shaded region =
36[tex]\pi[/tex]-72
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Secondly, we have to find the perimeter. This time, we have to find AC and measurement of arcAC and add them together to get the perimeter. To find AC we can use Pythagorean Theorem. To find the measurement of arcAC, we can take 1/4th of the circumference of the circle(This is the circle's perimeter). We know it's 1/4th because, like previously stated, 90 is 1/4th of 360, and this is a circle, so the measurement of arcAC is 1/4th of the circumference of the circle.
AC:
[tex]a^{2} +b^{2} =c^{2}[/tex]
The above is the Pythagorean theorem. We can substitute the side lengths of triangle ABC into this equation, where a and b are the legs and c is the hypotenuse.
12*12+12*12=[tex]AC^{2}[/tex]
144+144=[tex]AC^{2}[/tex]
2*144=[tex]AC^{2}[/tex]
Now we have to square root on both sides to get AC by itself.
AC=[tex]\sqrt{2*144}[/tex]
AC=[tex]\sqrt{2}[/tex] * [tex]\sqrt{144}[/tex]
We know the square root of 144 is 12, so we can simplify the equation to get
AC=12[tex]\sqrt{2}[/tex]
Measurement of arcAC:
See above for how we will calculate it.
2[tex]\pi[/tex]r= Area of a circle with radius r
We know the radius is 12 from before, but we need 1/4 of the circumference to get the measurement of arcAC.
2[tex]\pi[/tex]*12*1/4= Measurement of arcAC
6[tex]\pi[/tex]=Measurement of arcAC
Now to get the perimeter, we need to add these two values up
Perimeter of shaded region= AC+measurement of arcAC=12[tex]\sqrt{2}[/tex] + 6[tex]\pi[/tex]
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Have a great day!
Answer:
[tex]\sf A=\boxed{\sf 36\pi - 72}\; cm^2[/tex]
[tex]\sf P=\boxed{\sf 12\sqrt{2}+6\pi}\;cm[/tex]
Step-by-step explanation:
The given diagram shows quarter circle ABC, where AB and BC are the radii. The line segment connecting A and C is the hypotenuse of right triangle ABC.
Area of the shaded region
To find the area of the shaded region, subtract the area of right triangle ABC from the area of sector ABC:
Since the sector is a quarter circle, its area is a quarter of the area of a circle with radius 12 cm. The base and height of triangle ABC are the radii of the circle (12 cm). Therefore:
[tex]\sf \textsf{Area of shaded region}=\textsf{Area of sector ABC}-\textsf{Area of triangle ABC}\\\\\\\textsf{Area of shaded region}=\dfrac{\pi r^2}{4}-\left(\dfrac{1}{2} \cdot AB \cdot AC\right)\\\\\\\textsf{Area of shaded region}=\dfrac{\pi \cdot 12^2}{4}-\left(\dfrac{1}{2} \cdot 12 \cdot 12\right)\\\\\\\textsf{Area of shaded region}=\dfrac{144 \pi}{4}-\dfrac{144}{2}\\\\\\\textsf{Area of shaded region}=36\pi-72\; cm^2[/tex]
So, the area of the shaded region is:
[tex]\LARGE\boxed{\boxed{\sf Area=36\pi - 72\; cm^2}}[/tex]
[tex]\dotfill[/tex]
Perimeter of the shaded region
Since legs AB and BC of right triangle ABC are congruent, triangle ABC is a special 45-45-90 triangle. The sides of a 45-45-90 triangle are in the ratio 1 : 1 : √2, so the hypotenuse is √2 times the length of each leg. Since the legs of right triangle ABC both measure 12 cm:
[tex]\sf AC = 12\sqrt{2}\;cm[/tex]
As sector ABC is a quarter circle, the length of arc AC is a quarter of the circumference of a circle with radius 12 cm. The formula for the circumference of a circle with radius r is 2πr. Given that the radius of the circle is 12 cm:
[tex]\sf Arc\;AC=\dfrac{2\pi r}{4}=\dfrac{2\pi \cdot 12}{4}=6\pi\; cm[/tex]
The perimeter of the shaded region is the sum of line segment AC and arc AC. Therefore:
[tex]\sf Perimeter=Line\;segment\;AC+Arc\;AC\\\\Perimeter = 12\sqrt{2}+6\pi\; cm[/tex]
So, the perimeter of the shaded region is:
[tex]\LARGE\boxed{\boxed{\sf Perimeter=12\sqrt{2}+6\pi\; cm}}[/tex]
