c Subtract the area of the sector from the area of the right triangle we identified in Part A.
A
a About 549.78 cm^2
B
b About 74.2 cm^2
C
c About 25cm^2
Practice makes perfect
a When a line is tangent to a circle it will form a right angle with the radius of the circle. Let's add this information, along with the lengths of OE and XE to the diagram.
To find the area of the circle we need to know its radius. Observing the diagram, we see that the radius makes up a leg in a right triangle where the hypotenuse and second leg are known. Therefore, we can find the radius by using the Pythagorean Theorem.
We know that the radius is r= sqrt(175), so we can calculate the area of the circle.
A=π( sqrt(175))^2≈ 549.78 cm^2
b To find the area of the sector we have to know the sector's central angle. Since we know the hypotenuse and legs of the right triangle we identified in Part A, we can find this angle by using any of the trigonometric ratios — sine, cosine, or tangent.
When we know the central angle we have to multiply the area of the circle with the ratio of the central angle to 360^(∘).
&A=π(sqrt(175))^2* 48.59^(∘)/360^(∘)≈ 74.2 cm^2
c Let's highlight the region in our diagram.
In Part B we calculated the area of the sector bounded by OX and ON. To find the area of the region bounded by XE, NE, and NX we also have to calculate the area of △ EXO. Since this is a right triangle where both legs are known, we can find the area.
A=1/2(15)(sqrt(175))≈ 99.22 cm^2
Now we can calculate the region by subtracting the sector from the right triangle.
99.22-74.2≈ 25cm^2
