Question

Abcd is a cyclic quadrilateral such that angle a (4y+20) angle b=(3y-5) angle c=-4x and angle d=7x+5

Answer 1

Answer:

Step-by-step explanation:

Concept:

Opposite angles of a cyclic quadrilateral

are supplementary.

Given:

∠A = 4y+20

∠B = 3y-5

∠C = - 4x

∠D = 7x+5

since Opposite angles of a cyclic quadrilateral are supplementary,

we have

∠A+∠C = 180°

4y+20-4x = 180

-4x+4y = 160

-x+y = 40.........(1)

and

∠B+∠D = 180°

3y-5+7x+5 = 180

7x+3y= 180.....(2)

-7x+7y= 280 (multiplying (1) by 7)

10y = 460

y= 46

(1)gives

-x+46=40

-x= -6

x=6

Answer 2

Solution:

As we know that a cyclic quadrilateral is that in which sum of opposite angles are 180°.

here abcd is cyclic quadrilateral,so

angle a+ angle c= 180°

$4y + 20 - 4x = 180 \\ \\ 4y - 4x = 160 \\ \\ y - x = 40...eq1$
angle b+ angle d= 180°

$3y - 5 + 7x + 5 = 180 \\ \\ 3y +7x = 180...eq2$

substitute the value of y= 40+x from eq1 to eq 2

$3(40 + x) + 7x = 180 \\ \\ 120 + 3x + 7x = 180 \\ \\ 10x = 60 \\ \\ x = 6$
and
$y -6 = 40 \\ \\ y = 46$
so for x=6 and y= 46 abcd can only be a cyclic quadrilateral