Chapter 12, Problem 25CR

To determine

To calculate: The length of diagonal ET in given rectangle RECT in $\odot R$ .

Answer to Problem 25CR

The length of the diagonal ET in given rectangle RECT in $\odot R$ is $7\text{ units}$ .

Explanation of Solution

Given information: Given, $RT=5\text{ units}$ and $TQ=2\text{ units}$ .

Formula used:

Pythagoras Theorem,

  $h^2=b^2+p^2$

where h = hypotenuse of the right-angled triangle

b = base of the right-angled triangle

p = perpendicular of the right-angled triangle

Calculation:

From the given diagram we can easily observe that the radius of $\odot R$ ,

  $RQ=RT+TQ$

Substituting the given values of RT and TQ in above equation to get

  $\begin{array}{l}RQ=5+2\\ RQ=7\text{ units}\text{.}\end{array}$

Also, $RQ=RC=7\text{ units}$ (As both RQ and RC are the radius of $\odot R$ )

Now in $△RTC$ using Pythagoras theorem to have,

  $\begin{array}{l}R{C}^2=R{T}^2+T{C}^2\\ T{C}^2=R{C}^2-R{T}^2\end{array}$

Taking square roots of the terms on both sides

  $TC=\sqrt{R{C}^2-R{T}^2}$ ………………………. (i)

Substituting the given value $RT=5\text{ units}$ and the calculated value $RC=7\text{ units}$ in equation (i)

  $\begin{array}{l}TC=\sqrt{7^2-5^2}\\ TC=\sqrt{49-25}\\ TC=\sqrt{24}\text{ units}\\ TC=4.899\text{ units}\text{.}\end{array}$

We know that the opposite sides of any rectangle are equal. So, in rectangle RECT

  $RE=TC=\sqrt{24}\text{ units = }4.899\text{ units}$

Now in $△ERT$ using Pythagoras theorem to have,

  $E{T}^2=E{R}^2+R{T}^2$

Taking square root of the terms on both sides

  $ET=\sqrt{E{R}^2+R{T}^2}$ ……………………… (ii)

Substituting the given value $RT=5\text{ units}$ and the calculated value $ER=\sqrt{24}\text{ units}$ in equation (ii) to have,

  $\begin{array}{l}ET=\sqrt{{\left(\sqrt{24}\right)}^2+5^2}\\ ET=\sqrt{24+25}\\ ET=\sqrt{49}\\ ET=7\text{ units}\text{.}\end{array}$

The length of the diagonal ET in given rectangle RECT in $\odot R$ is $7\text{ units}$ .