Question

SOLVE FOR 8 POINTS
In figure 11.36, ABCD is a tripezium. If
$x = \frac{4}{3} y$
$y = \frac{3}{8}z$

,then find the value of
$x$

​

Answer 1

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Given :

$\sf{x = \dfrac{4}{3} y}$

$\sf{y = \dfrac{3}{8}z}$

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To Find :

• Value of x

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Solution :

$\sf{x \: = \: \dfrac{4}{3}y} \\ \\ \implies {\sf{y \: = \: \dfrac{3}{4}x}}$

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$\sf{y \: = \: \dfrac{3}{8}z} \\ \\ \implies {\sf{z \: = \: \dfrac{8}{3}y}}$

Put value of y

$\implies {\sf{z \: = \: \dfrac{8}{3} \: \times \: \dfrac{3}{4}x}} \\ \\ \implies {\sf{z \: = \: 2x}}$

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Use angle sum property of triangle in ΔBCD

$\implies {\sf{\angle BCD \: + \: \angle DBC \: + \: \angle BDC \: = \: 180 }} \\ \\ \implies {\sf{x \: + \: y \: + \: z \: = \: 180}} \\ \\ \implies {\sf{x \: = \: \dfrac{3}{4}x \: + \: 2x \: = \: 180}} \\ \\ \implies {\sf{\dfrac{4x\: + \: 3x \: + \: 8x}{4} \: = \: 180}} \\ \\ \implies {\sf{\dfrac{15x}{4} \: = \: 180}} \\ \\ \implies {\sf{15x \: = \: 180 \: \times \: 4}} \\ \\ \implies {\sf{15x \: = \: 720}} \\ \\ \implies {\sf{x \: = \: \dfrac{720}{15}}}\\ \\ \implies {\sf{x \: = \: 48}}$

Value of x is 48°

Answer 2

Solution :-

We have to find the value of x

So, we need to find values of y and z in terms of x

1) x = 4y/3

⇒ 3x/4 = y

⇒ y = 3x/4

2) y = 3z/8

⇒ 8y/3 = z

⇒ z = 8y/3

Substituting y = 3x/4

⇒ z = 8/3 * (3x/4)

⇒ z = 2x

Now In trapezium ABCD

AB || CD and BD is transvervsal

⇒ ∠ABD = ∠BDC = x [ Alternate angles are equal ]

Consider ΔBDC

By Angle sum property

∠BDC + ∠DBC + ∠BCD = 180°

⇒ x + y + z = 180°

Substituting y = 3x/4 and z = 2x

⇒ x + 3x/4 + 2x = 180

⇒ (4x + 3x + 8x) /4 = 180

⇒ 15x/4 = 180

⇒ 15x = 180 * 4

⇒ x = (180 * 4)/15

⇒ x = 12 * 4 = 48°

Therefore the value of x is 48°.