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A rectangle is inscribed in a circle with a diameter lying along the line5y=x-7 If the two adjacent vertices of the rectangle are(-7,4) and then the area of the rectangle (in sq units) is 

 

Option: 1

20 square units

 

 


Option: 2

25 square units

 


Option: 3

30 square units

 


Option: 4

50 square units 


Answers (1)

best_answer

Slope of the diameter =


\begin{aligned} & (y 2-y 1) /(x 2-x 1) \\ = & (2-4) /(3-(-7)) \\ = & -2 / 10 \\ = & -1 / 5 \end{aligned}

The midpoint of the diameter is the center of the circle. 

Midpoint - 


\begin{aligned} & =((x 1+x 2) / 2,(y 1+y 2) / 2) \\ = & ((-7+3) / 2,(4+2) / 2) \\ = & (-2,3) \end{aligned}

So, the center of the circle is (-2,3) and the radius is half the distance between the center and one of the vertices of the rectangle:

Radius - 

\begin{aligned} & \left.=\sqrt{(}(-2-(-7))^2+(3-4)^2\right) / 2 \\ = & \sqrt{(25+1) / 2} \\ = & \sqrt{(26) / 2} \end{aligned}

The Pythagorean theorem to find the length and width of the rectangle:

length = distance between (-7,4)and(3,4)\begin{aligned} & \left.=\sqrt{(}(3-(-7))^2+(2-4)^2\right) \\ = & \sqrt{(100)} \\ = & 10 \end{aligned}

width = distance between (-7,4) and a point on the circle that is on the same vertical line as (3,2)= distance between (-7,4) and(2,4)

Therefore, the area of the rectangle is:

area = length * width

10*5

50square units.

So the area of the rectangle is 50 square units.

 

Posted by

Deependra Verma

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