CHAPTER $4$ . CTRCULAR MEASURE $\newline$ The diagram shows a circle with centre $A$ and radius $r$ . Diameters $C A D$ and $B A E$ are perpendicular to each other. A larger circle has centre $B$ and passes through $C$ and $D$ . $\newline$ (i) Show that the radius of the larger circle is $r \sqrt{ } 2$ . $\newline$ [I] $\newline$ (ii) Find the area of the shaded region in terms of $r$ . $\newline$ [ $6$ ]

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Algebra 2

Find Coordinate on Unit Circle

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Q. CHAPTER

4

. CTRCULAR MEASURE

\newline

The diagram shows a circle with centre

A

and radius

r

. Diameters

C A D

and

B A E

are perpendicular to each other. A larger circle has centre

B

and passes through

C

and

D

\newline

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

r \sqrt{ } 2

\newline

[I]

\newline

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

r

\newline

[

6

]

Triangle Properties: (i) Since $CAD$ and $BAE$ are perpendicular diameters, triangle $ABC$ is a right triangle with $AC$ and $BC$ as its legs, and $AB$ as its hypotenuse.
Pythagorean Theorem: Using the Pythagorean theorem, $AB^2 = AC^2 + BC^2$ . Since $AC$ and $BC$ are both radii of the smaller circle, $AB^2 = r^2 + r^2$ .
Radius Calculation: Simplify to get $AB^2 = 2r^2$ , so $AB = \sqrt{2r^2} = r\sqrt{2}$ . This is the radius of the larger circle.
Area of Larger Circle: (ii) The area of the larger circle is $\pi*(\text{radius of larger circle})^2$ . Substituting the radius we found, the area is $\pi*(r*\sqrt{2})^2$ .
Subtracting Areas: Simplify the area of the larger circle to get $\pi*(r^2*2) = 2\pi r^2$ .
Final Shaded Area Calculation: The area of the smaller circle is $\pi r^2$ . To find the shaded area, subtract the area of the smaller circle from the area of the larger circle.
Final Shaded Area Calculation: The area of the smaller circle is $\pi r^2$ . To find the shaded area, subtract the area of the smaller circle from the area of the larger circle. Shaded area = Area of larger circle - Area of smaller circle = $2\pi r^2 - \pi r^2$ .
Final Shaded Area Calculation: The area of the smaller circle is $\pi r^2$ . To find the shaded area, subtract the area of the smaller circle from the area of the larger circle. Shaded area = Area of larger circle - Area of smaller circle = $2\pi r^2 - \pi r^2$ . Simplify the shaded area to get $\pi r^2$ . This is the area of the shaded region in terms of $r$ .