A rectangle is inscribed in a circle with a diameter lying along the line 3y=x+7.

Question

A rectangle is inscribed in a circle with a diameter lying along the line $3y=x+7$. If the two adjacent vertices of the rectangle are $(-8,5)$ and $(6,5)$, then the area of the rectangle (in sq. units) is:

Accepted Answer

Distance beetween point A and B is  = 14 unit

As y coordinate of point A(-8, 5) and B(6, 5) are same. So line AB is parallel to the x axis.

Point O is the center of the circle and OP is the perpendicular to the line AB. And P is the mis point of line AB.

$$ \therefore $$ Point P = $$\left( {{{ - 8 + 6} \over 2},{{5 + 5} \over 2}} \right)$$ = (-1, 5)

As OP is perpendicular to the line AB and AB is parallel to the x axis then OP is parallel to the y axis. So x coordinate of line OP is constant.

$$ \therefore $$ x coordinate of O  = -1

And point O is lies on the line 3y = x + 7

$$ \therefore $$ 3y = -1 + 7

$$ \Rightarrow $$ y = 2

$$ \therefore $$ Center O = (-1, 2)

$$ \therefore $$ OP = 3

Then BC = 2OP = 6

$$ \therefore $$ Area of rectangle = (AB)(BC)

= 14 $$ \times $$ 6

= 84

Solutions · 1