Name: Cross Multiplication Method: Illustration
Uploaded: 2019-07-19
Duration: 7 min 27 s
Description: This tutorial will show you many examples of solving linear equations in two variables using the cross-multiplication method with a clear step-by-step explan...

Question

A B C D

is a cyclic quadrilateral such that

∠ A = (4 y + 20) °, ∠ B = (3 y - 5) °, ∠ C = (4 x) °

and

∠ D = (7 x + 5) °

. Find the four angles.

Accepted Answer

We know that the sum of the opposite angles of cyclic quadrilateral is $180 °$ . In the cyclic quadrilateral $∠ A = (4 y + 20) °, ∠ B = (3 y - 5) °, ∠ C = (4 x) °$ angles $A$ and $C$ and angles $B$ and $D$ are pair of opposite angles
Therefore,
$∠ A + ∠ C = 180 °$ and $∠ B + ∠ D = 180 °$
By substituting $∠ A = (4 y + 20) °$ and $∠ C = (4 x) °$ , we get
$4 y + 20 ° + 4 x = 180 °$
$4 x + 4 y + 20 ° = 180 °$
$4 x + 4 y = 180 ° - 20 °$
$4 x + 4 y = 160 °$
Divide both sides of the equation by $4$ , we get
$x + y = 40 ° \dots \dots (i)$
$∠ B + ∠ D = 180 °$
By substituting $∠ B = (3 y - 5) °$ and $∠ D = (7 x + 5) °$ , we get
$3 y - 5 + 7 x + 5 = 180 °$
$7 x + 3 y = 180 ° \dots \dots (ii)$

By multiplying equation (i) by $3$ , we get
$3 x + 3 y = 120 °$ …(iii)
By subtracting equation (iii) from (ii) we get
$\begin{matrix} 7 x + & 3 y = & 180 \\ - 3 x - & 3 y = - & 120 \end{matrix} \bar{4 x + 0 y = 60}$
$x = \frac{60}{4}$
$x = 15$
By substituting $x = 15 °$ in equation (i) we get
$\Rightarrow x + y = 40$
$\Rightarrow 15 + y = 40$
$\Rightarrow y = 40 - 15$
$\Rightarrow y = 25$
The angles of a cyclic quadrilateral are
$∠ A = 4 y + 20$
$= 4 \times 25 + 20$
$= 100 + 20$
$= 120 °$
$∠ B = 3 y - 5$
$= 3 \times 25 - 5$
$= 75 - 5$
$= 70 °$
$∠ C = 4 x$
$= 4 (15)$
$= 60 °$
$∠ D = 7 x + 5$
$= 7 \times 15 + 5$
$= 105 + 5$
$= 110 °$
Hence, the angles of quadrilateral are $∠ A = 120 °, ∠ B = 70 °, ∠ C = 60 °, ∠ D = 110 °$ .

$A B C D$ is a cyclic quadrilateral such that $∠ A = (4 y + 20) °, ∠ B = (3 y - 5) °, ∠ C = (4 x) °$ and $∠ D = (7 x + 5) °$ . Find the four angles.

Important Questions on Pair of Linear Equations in Two Variables

Learn and Prepare for any exam you want !

ABCD is a cyclic quadrilateral such that ∠A=4y+20°, ∠B=3y-5°, ∠C=4x° and ∠D=7x+5°. Find the four angles.

Important Questions on Pair of Linear Equations in Two Variables

Learn and Prepare for any exam you want !

$A B C D$ is a cyclic quadrilateral such that $∠ A = (4 y + 20) °, ∠ B = (3 y - 5) °, ∠ C = (4 x) °$ and $∠ D = (7 x + 5) °$ . Find the four angles.