Question

Plsssss help.

A,B,C and D lie on a circle, center O, radius 8cm. AB and CD are tangents to a circle, center O, radius 4cm.
ABCD is a rectangle. a) calculate the distance AE
b) calculate the shaded area

Answer 1

Answer:

a)  $AE=4\sqrt{3}$

b) $A_s=16(4\sqrt{3}-\pi)\ cm^2$

Step-by-step explanation:

Lines in a Circle

(a)

Points AEO form a right triangle, which hypotenuse is the radius of the larger circle (8 cm) and one of its legs is the radius of the smaller circle (4 cm).

Thus, the distance AE is calculated by using the Pythagora's Theorem:

$AE^2=8^2-4^2$

$AE^2=64-16=48$

$AE=\sqrt{48}$

Since 48=16*3:

$AE=\sqrt{16}\sqrt{3}$

$\boxed{AE=4\sqrt{3}}$

(b)

The shaded area is the area of the rectangle ABCD minus the area of the smaller circle:

$A_s=A_r-A_c$

The rectangle has a length of twice AE:

$L=8\sqrt{3}\ cm$

And a width equal to the diameter of the smaller circle:

$W=8\ cm$

The area of the rectangle is:

$A_r=8\sqrt{3}\ cm*8\ cm$

$A_r=64\sqrt{3}\ cm^2$

The area of the smaller circle is:

$A_c=\pi\ r^2$

$A_c=\pi\ 4^2\ cm^2$

$A_c=16\pi\ cm^2$

Thus, the shaded area is:

$A_s=(64\sqrt{3}-16\pi)\ cm^2$

Factoring:

$\boxed{A_s=16(4\sqrt{3}-\pi)\ cm^2}$

Answer 2

a. the length of AE is: 6.9 cm

b. the shaded area is: 60.13 sq. cm.

Recall:

• Area of rectangle = length × width
• Area of circle = πr²

a. Find AE:

Considering right triangle AOE, apply the Pythagorean Theorem to find AE, given that,

EO = 4 cm

AO = 8 cm

Thus:

AE = √(AO² - EO²)

AE = √(8² - 4²)

AE = 6.9 cm

b. Shaded Area = Area of rectangle - area of circle

Area of rectangle = length × width = 2(6.9) × 2(4) = 110.4 sq. cm

Area of circle = π(4)² = 50.27 sq. cm

Shaded Area = 110.4 - 50.27 = 60.13 sq. cm.

Learn more about shaded area on:

https://brainly.com/question/13515841