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The diagram shou a circle with centre
A and radius
r. Diameters
CAD and
BAE are perpendicalar to each other. A larger circte has centre
B and paswes through
C and
D.
(i) Show that the radius of the larger circle is
rsqrt()2.
iti
(ii) Find the area of the shaded region in terms of
r.
|6]

The diagram shou a circle with centre $A$ and radius $r$ . Diameters $C A D$ and $B A E$ are perpendicalar to each other. A larger circte has centre $B$ and paswes through $C$ and $D$ . $\newline$ (i) Show that the radius of the larger circle is $r \sqrt{ } 2$ . $\newline$ iti $\newline$ (ii) Find the area of the shaded region in terms of $r$ . $\newline$ | $6$ ]

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Area of quadrilaterals and triangles: word problems

Full solution

Q. The diagram shou a circle with centre

A

and radius

r

. Diameters

C A D

and

B A E

are perpendicalar to each other. A larger circte has centre

B

and paswes through

C

and

D

.

\newline

(i) Show that the radius of the larger circle is

r \sqrt{ } 2

.

\newline

iti

\newline

(ii) Find the area of the shaded region in terms of

r

.

\newline

|

6

]

Triangle $ACD$ Right Triangle: Since $CAD$ and $BAE$ are perpendicular diameters, triangle $ACD$ is a right triangle with legs of length $r$ .
Calculate Larger Circle Radius: Using the Pythagorean theorem, the hypotenuse of triangle ACD, which is the radius of the larger circle, is found by $r^2 + r^2 = (\text{radius of larger circle})^2$ .
Find Area of Larger Circle: Calculating the radius of the larger circle gives $(\text{radius of larger circle})^2 = 2r^2$ , so the radius of the larger circle is $r\sqrt{2}$ .
Subtract Area of Smaller Circle: For the area of the shaded region, we first find the area of the larger circle using the formula $A = \pi(\text{radius})^2$ .
Calculate Shaded Area: Substituting the radius of the larger circle, we get $A = \pi(r\sqrt{2})^2$ .
Calculate Shaded Area: Substituting the radius of the larger circle, we get $A = \pi(r\sqrt{2})^2$ . Expanding the equation, we get $A = \pi(2r^2)$ .
Calculate Shaded Area: Substituting the radius of the larger circle, we get $A = \pi(r\sqrt{2})^2$ . Expanding the equation, we get $A = \pi(2r^2)$ . Now, we need to subtract the area of the smaller circle from the area of the larger circle to find the shaded area.
Calculate Shaded Area: Substituting the radius of the larger circle, we get $A = \pi(r\sqrt{2})^2$ . Expanding the equation, we get $A = \pi(2r^2)$ . Now, we need to subtract the area of the smaller circle from the area of the larger circle to find the shaded area. The area of the smaller circle is $A = \pi r^2$ .
Calculate Shaded Area: Substituting the radius of the larger circle, we get $A = \pi(r\sqrt{2})^2$ .Expanding the equation, we get $A = \pi(2r^2)$ .Now, we need to subtract the area of the smaller circle from the area of the larger circle to find the shaded area.The area of the smaller circle is $A = \pi r^2$ .Subtracting the area of the smaller circle from the area of the larger circle gives us the shaded area: $\pi(2r^2) - \pi r^2$ .
Calculate Shaded Area: Substituting the radius of the larger circle, we get $A = \pi(r\sqrt{2})^2$ .Expanding the equation, we get $A = \pi(2r^2)$ .Now, we need to subtract the area of the smaller circle from the area of the larger circle to find the shaded area.The area of the smaller circle is $A = \pi r^2$ .Subtracting the area of the smaller circle from the area of the larger circle gives us the shaded area: $\pi(2r^2) - \pi r^2$ .Calculating the shaded area, we get the shaded area = $\pi r^2$ .

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