Question

the radius of the circle is 4 cm and the measure of the central angle is 90°. the area of the sector with a central angle measuring 90° and radius of length 4 cm is π cm². the triangle in the sector is . the area of the triangle is cm². the area of the segment of the circle is (4π - ) cm². 4 8 16

Explanation:

Step1: Calculate sector area

The formula for the area of a sector of a circle is $A=\frac{\theta}{360^{\circ}}\times\pi r^{2}$, where $\theta$ is the central - angle and $r$ is the radius. Given $\theta = 90^{\circ}$ and $r = 4$ cm. Then $A=\frac{90^{\circ}}{360^{\circ}}\times\pi\times4^{2}=\frac{1}{4}\times16\pi = 4\pi$ cm².

Step2: Identify the triangle type

The triangle in the sector has two sides equal to the radius of the circle ($r = 4$ cm) and the included - angle is $90^{\circ}$, so it is a right - isosceles triangle.

Step3: Calculate triangle area

The formula for the area of a right - triangle is $A=\frac{1}{2}ab$, where $a$ and $b$ are the two legs of the right - triangle. Here $a = b=4$ cm, so $A=\frac{1}{2}\times4\times4 = 8$ cm².

Step4: Calculate segment area

The area of the segment is the area of the sector minus the area of the triangle. So the area of the segment is $(4\pi−8)$ cm².

Answer:

The area of the sector with a central angle measuring $90^{\circ}$ and radius of length 4 cm is $4\pi$ cm².
The triangle in the sector is a right - isosceles triangle.
The area of the triangle is $8$ cm².
The area of the segment of the circle is $(4\pi - 8)$ cm².