Question

The diameter of circumcircle of a rectangle is 10cm and breadth is 6cm. Find its length.
[a] 6cm
[b] 5cm
[c] 8cm
[d] None of these.

Answer 1

Hint: Use the fact that if ABCD is a rectangle, then AC is the diameter of the circumcircle of the rectangle. Hence find the length of AC. Use the fact that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides. Hence find the length of the side AD. Hence determine the length of the rectangle.
Complete step by step solution:

Given: ABCD is a rectangle. The diameter of the circumcircle of the rectangle ABCD is 10cm and AB = 6cm
To determine: AD
Construction: Join AC.
We know that if the angle in a segment is a right angle, then the segment is a semicircle. Since ABCD is a rectangle, angle ADC is a right angle, and hence AC is the diameter of the circle. Hence, we have AC = 10cm
Now, we know that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides. This is known as Pythagoras theorem.
Hence by Pythagoras theorem in triangle ABC, we have
$A{{B}^{2}}+B{{C}^{2}}=A{{C}^{2}}$
Substituting AB= 6cm and AC = 10cm, we have
$\begin{align}
  & {{6}^{2}}+B{{C}^{2}}={{10}^{2}} \\
 & \Rightarrow B{{C}^{2}}+36=100 \\
\end{align}$
Subtracting 36 from both sides, we get
$B{{C}^{2}}=64$
Hence, we have BC = 8cm
Hence the length of the rectangle is 8cm.
Hence option [c] is correct.
Note: Alternative solution:

Let the length of the rectangle be l
Hence we have
Area of the rectangle $=lb$
Now, we have AE = 5cm, BE =5cm ad AB = 6cm
Hence in triangle ABE, we have
a=5, b=5 and c= 6
Hence, we have
$s=\dfrac{a+b+c}{2}=\dfrac{5+5+6}{2}=8$
Now, by Heron’s formula, we have
$\Delta =\sqrt{s\left( s-a \right)\left( s-b \right)\left( s-c \right)}$
Hence, we have
$ar\left( \Delta ABE \right)=\sqrt{8\left( 8-5 \right)\left( 8-5 \right)\left( 8-6 \right)}=12$
Now, we have
$ar\left( ABCD \right)=4ar\left( \Delta ABE \right)=4\times 12=48$
Hence, we have
$lb=48$
Here b = 6
Hence, we have
$6l=48$
Dividing both sides by 6, we get
$l=8$
Hence, the length of the rectangle is 8cm.