Question

find the measure of angle a.
11)
a
x + 46
x + 56
a) 50° b) 40°
c) 60° d) 85°
find the circumference of each circle.

1. area = 2209π cm²

a) 76π cm b) 98π cm
c) 94π cm d) 56π cm

Explanation:

Question 11 (Finding measure of angle A)

Step1: Recall triangle angle sum

In a right triangle, the two non - right angles sum to \(90^{\circ}\). So, \((x + 56)+(x + 46)=90\).

Step2: Solve for x

Combine like terms: \(2x+102 = 90\). Subtract 102 from both sides: \(2x=90 - 102=- 12\)? Wait, that can't be right. Wait, no, the right angle is \(90^{\circ}\), and the sum of all angles in a triangle is \(180^{\circ}\). So the two acute angles sum to \(90^{\circ}\). Wait, maybe I misread the angles. Wait, angle at the right angle is \(90^{\circ}\), so angle A and the angle with measure \(x + 56\) are the two acute angles. So \(m\angle A+(x + 56)=90\), and \(m\angle A=x + 46\)? Wait, no, let's re - express. Let the two acute angles be \(A\) (with measure \(x + 46\)) and the other acute angle (with measure \(x + 56\)). Then \((x + 46)+(x + 56)=90\).
\(2x+102 = 90\)
\(2x=90 - 102=-12\), this is wrong. Wait, no, maybe the right angle is \(90^{\circ}\), and the sum of all three angles is \(180^{\circ}\). So \((x + 56)+(x + 46)+90 = 180\)
\(2x+102+90=180\)
\(2x + 192=180\)
\(2x=180 - 192=-12\), still wrong. Wait, maybe the labels are different. Wait, the right angle is at the bottom right. So angle at the bottom left is \(x + 56\), angle at A is \(x + 46\), and the right angle is \(90^{\circ}\). So sum of angles: \((x + 56)+(x + 46)+90=180\)
\(2x+102 + 90=180\)
\(2x+192 = 180\)
\(2x=- 12\), \(x=-6\). Then angle A is \(x + 46=-6 + 46 = 40^{\circ}\)

Step1: Recall area formula of a circle

The area of a circle is given by \(A=\pi r^{2}\), where \(A = 2209\pi\space cm^{2}\). So \(\pi r^{2}=2209\pi\). Divide both sides by \(\pi\): \(r^{2}=2209\).

Step2: Find the radius

Take the square root of both sides: \(r=\sqrt{2209}=47\space cm\)? Wait, no, \(\sqrt{2209} = 47\)? Wait, \(47\times47 = 2209\)? No, \(47\times47=2209\)? Wait, \(46\times46 = 2116\), \(47\times47 = 2209\), \(48\times48=2304\). Wait, but the options for circumference are \(76\pi\), \(98\pi\), etc. Wait, circumference \(C = 2\pi r\). If \(r = 38\), then \(C=76\pi\). Wait, maybe I made a mistake in calculating \(r\). Wait, \(2209 = 47^{2}\)? No, \(47^{2}=2209\), but \(2\times47 = 94\), no. Wait, \(38^{2}=1444\), \(43^{2}=1849\), \(47^{2}=2209\), \(38\times2 = 76\). Wait, maybe the area is \(2209\pi\), so \(r^{2}=2209\), \(r = 47\)? No, that can't be. Wait, maybe the area is \(1369\pi\) (since \(37^{2}=1369\), \(38^{2}=1444\), \(43^{2}=1849\), \(47^{2}=2209\), \(38\times2=76\)). Wait, maybe there is a typo, but assuming the area is \(1369\pi\), \(r = 38\), \(C = 2\pi r=76\pi\). Alternatively, if we consider that \(2209 = 47^{2}\), but \(2\times47 = 94\), which is option C. Wait, no, let's re - check. The area formula \(A=\pi r^{2}\), so \(r=\sqrt{\frac{A}{\pi}}\). Given \(A = 2209\pi\), \(r=\sqrt{2209}=47\). Then circumference \(C = 2\pi r=2\pi\times47 = 94\pi\)? But option C is \(94\pi\). Wait, earlier mistake: \(47\times2 = 94\). So:

Step1: Find radius from area

Given \(A=\pi r^{2}=2209\pi\), divide both sides by \(\pi\): \(r^{2}=2209\), so \(r = \sqrt{2209}=47\space cm\).

Step2: Calculate circumference

Circumference \(C = 2\pi r=2\pi\times47 = 94\pi\space cm\)

Answer:

B) \(40^{\circ}\)

Question 12 (Finding circumference of the circle)