a rectangle with one side of length 4cm is inscribed in a circle of diameter 5cm find the area of the rectangle.
it is from the chapter circle of understanding ICSE Mathematics class 10 by M.L Aggarwal.
Let figure out the situation.
Inside the circle there are four corners of the rectangle touching the circumference of the circle. lets name this corners A,B,C and D respectively.
Now AB is a length and BC is a width of the rectangle. If we consider ABC it is a right triangle and AC is hypotenuse.AC goes through the center of the circle forming a diameter.
AB = 4
AC = 5
Using Pythagoras law
`AC^2 = AB^2+BC^2`
`5^2 = 4^2+BC^2`
`BC = 3`
So the rectangle has its legs as 4 and 3.
Area of rectangle = AB*BC = 4*3 = 12
Area of rectangle is `12cm^2`
The answer is clear .
since the rectangle is inscribed in the circle, the diameter of the circle = the diagonal of the rectangle=5cm
now the diagonal of the rectangle is the hypotenuse and let x be the other side of the rectangle
Using pythagoras theorem,
(x)^2 + (4)^2 = (5)^2
hence, x = 3 cm