Ex.3.7 Q8 Pair of Linear Equations in Two Variables Solution - NCERT Maths Class 10

\(ABCD\) is a cyclic quadrilateral finds the angles of the cyclic quadrilateral.

Text Solution

What is Known?

Measurement of the angles of the cyclic quadrilateral in terms of \(x\) and \(y.\)

What is Unknown?

Measurement of the angles of the cyclic quadrilateral.

Reasoning:

Pairs of opposite angles of a cyclic quadrilateral are supplementary.

Steps:

We know that the sum of the measures of opposite angles in a cyclic quadrilateral is \(180^\circ.\)

Therefore,

\[\begin{align}\angle A + \angle C &= {180^\circ}\\\left( {4y + 20} \right) + \left( { - 4x} \right) &= 180\\
4y + 20 - 4x &= 180\\ - 4\left( {x - y} \right)& = 160\\x - y &=  - 40 \qquad \left( 1 \right)\end{align}\]

And

\[\begin{align}\angle B + \angle D &= {180^\circ}\\\left( {3y - 5} \right) + \left( { - 7x + 5} \right) &= 180\\
3y - 5 - 7x + 5 &= 180\\ - 7x + 3y &= 180\\7x - 3y &=  - 180 \qquad \left( 2 \right)\end{align}\]

Multiplying equation \((1)\) by \(3,\) we obtain

\[3x - 3y =  - 120\qquad \left( 3 \right)\]

Subtracting equation \((3)\) from equation \((2),\) we obtain

\[\begin{align}4x &= - 60\\x &= - 15\end{align}\]

Substituting \(x = - 15\) in equation \((1),\) we obtain

\[\begin{align} - 15 - y &= - 40\\y &= 25\end{align}\]

Therefore,

\[\begin{align}\angle A &= 4 \times 25 + 20 = {120^\circ}\\\angle B &= 3 \times 25 - 5 = {70^\circ}\\
\angle C &= - 4 \times \left( { - 15} \right) = {60^\circ}\\\angle D &= - 7 \times \left( { - 15} \right) + 5 = {110^\circ}\end{align}\]