Question

1. the area of a circle is 111.6 square centimeters. what is the diameter? use 3.1 for π. 12 square centimeters 2. find the missing angle. 37° 62° x° 3. find the area of the shaded region. use 3 for π. 10 m 5 m 4. triangle abc is a right isosceles triangle. what are the measures of the three angles?

Explanation:

Step1: Find radius from area formula

Given $A = \pi r^{2}$, so $r=\sqrt{\frac{A}{\pi}}$. Substituting $A = 111.6$ and $\pi=3.1$, we get $r=\sqrt{\frac{111.6}{3.1}}=\sqrt{36}=6$ cm.

Step2: Calculate diameter

$d = 2r$, so $d=2\times6 = 12$ cm.

Step3: Find exterior - angle for second problem

The sum of interior angles of a triangle is $180^{\circ}$. The non - adjacent interior angles sum is $37^{\circ}+62^{\circ}=99^{\circ}$. By exterior - angle property, $x = 99^{\circ}$.

Step4: Calculate area for third problem

Area of rectangle $A_{rect}=10\times5 = 50$ m². Radius of circle $r=\frac{5}{2}$ m, area of circle $A_{circle}=3\times(\frac{5}{2})^{2}=3\times\frac{25}{4}=\frac{75}{4}=18.75$ m². Shaded area $A = 50 - 18.75=31.25$ m².

Step5: Determine angles for fourth problem

In right isosceles triangle, one angle is $90^{\circ}$. The other two equal angles sum to $180 - 90=90^{\circ}$, so each is $45^{\circ}$. Angles are $90^{\circ},45^{\circ},45^{\circ}$.

Answer:

1. 12 cm
2. $99^{\circ}$
3. $31.25$ m²
4. $90^{\circ},45^{\circ},45^{\circ}$