Question

In figure, if AB || DC, then find the value ofx.
{OD= 3 cm, OC = (x - 5) cm
OA = (3x – 19) cm, OB =(x – 3) cm}
X-5
X-3
33-19​

Answer 1

Question:

In figure, if AB || DC, then find the value of x.

{OD= 3 cm, OC = (x - 5) cm

OA = (3x – 19) cm, OB = (x – 3) cm}

Answer:

x = 8 or 9

Note:

• If two lines intersect at a point, then the vertically opposite angles are equal.

• A transversal is a line which cuts two lines at two distinct points in the same plane.

• If a transversal cuts two parallel lines, then the alternative interior angles are equal.

• If the crossponding sides of two triangles are proportional then both the triangles are said to be similar triangles.

• If two triangles are similar then their crossponding angles are equal.

Solution:

Given:

AB || DC

OD= 3 cm

OC = (x - 5) cm

OA = (3x - 19) cm

OB = (x- 3) cm

Since,

AB || DC and DB is transversal ,

Thus ,

Angle CDO = Angle OBA

{Alternate interior angles are equal}

Again,

AB || DC and CA is transversal ,

Thus ,

Angle DCO = Angle OAB

{Alternate interior angles are equal}

Also,

Angle COD = Angle AOB

{Vertically opposite angles are equal}

Now,

In ∆COD and ∆AOB

Angle CDO = Angle OBA

Angle DCO = Angle OAB

Angle COD = Angle AOB

Thus,

∆COD and ∆AOB are similar triangles.

{By A.A.A. similarity property of triangles}

Since ,

∆COD and ∆AOB are similar triangles.

Thus,

The crossponding sides of ∆COD and ∆AOB must be proportional.

ie; CO/OA = DO/OB = CD/AB

Thus;

=> CO/OA = DO/OB

=> (x-5)/(3x-19) = 3/(x-3)

=> (x-5)•(x-3) = 3•(3x-19)

=> x^2 - 3x - 5x + 15 = 9x - 57

=> x^2 - 8x + 15 = 9x - 57

=> x^2 - 8x - 9x + 15 + 57 = 0

=> x^2 - 8x - 9x + 72 = 0

=> x(x - 8) - 9(x - 8) = 0

=> (x - 8)(x - 9) = 0

=> x = 8 , 9

Hence,

The required values of x are 8 and 9 .

Answer 2

Given :----

• OD = 3cm
• OC = (x-5)cm
• OA = (3x-19)cm
• OB = (x -3) cm
• AB is parallel to DC ...

Question :------

• Find the value of X ... ?

Solution :-----

since it is given that ,, AB is parallel to DC ,

That means ,, ABCD is s Trapezium ...

And we know that,,,

$\textbf{The diagonals of a trapezium divide} \\ \textbf{each other proportionally.}$

Hence,

$\large\boxed{\bold{ \frac{DO} {OB} = \frac{OC}{AO}}}$

Putting values we get,,,,

$\frac{3}{x - 3} = \frac{x - 5}{3x - 19} \\ \\ \red\leadsto \: 9x - 57 = {x}^{2} - 5x - 3x + 15 \\ \\ \red\leadsto \: {x}^{2} - 17x + 72 = 0 \\ \\ \red\leadsto {x}^{2} - 8x - 9x + 72 = 0 \\ \\ \red\leadsto \: x(x - 8) - 9(x - 8) = 0 \\ \\ \red\leadsto(x - 8)(x - 9) = 0 \\ \\ \large\boxed{\bold{x = 8 , 9 }}$

Hence , value of x can be 8 and 9 both possible.....

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$\star \textbf{extra brainly knowledge} \star$

The properties of the diagonals of a trapezium are:------

1. The diagonals of a trapezium are unequal.

2. The diagonals of an isosceles trapezium are equal.

3. The point of intersection of the diagonals of an isosceles trapezium lies midway between the parallel sides.

4. The point of intersection of the diagonals of an isosceles trapezium lie on the diameter of the circumscribing circle.

5. point of intersection of the diagonals of an isosceles trapezium is the centre of the circumscribing circle.

6. The diagonals of an isosceles trapezium form two pairs of congruent triangles - one pair being obtuse triangles and the other pair, acute triangles.

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$\large\underline\textbf{Hope it Helps You.}$