Question

In the given figure, if AB || CD, find the value of x.​

Answer 1

$\large\underline{\sf{Solution-}}$

It is given that, in a quadrilateral ABCD, AB || CD

So, ABCD is a trapezium.

Now,

Consider,

$\rm :\longmapsto\:\boxed{ \tt{ \: \triangle \:AOB \: and \: \triangle \:COD \: }}$

$\rm :\longmapsto\: \angle \:OAB \: = \:\angle \: OCD \: \: \{alternate \: interior \: angles \}$

$\rm :\longmapsto\: \angle \:OBA \: = \:\angle \: ODC \: \: \{alternate \: interior \: angles \}$

Therefore,

$\rm :\longmapsto\:\triangle \:AOB \: \sim \: \triangle \:COD \: \: \: \: \: \: \{AA \}$

So,

$\rm :\longmapsto\:\dfrac{OA}{OC} = \dfrac{OB}{OD}$

$\rm :\longmapsto\:\dfrac{x + 3}{x + 5} = \dfrac{x - 2}{x - 1}$

$\rm :\longmapsto\:(x + 5)(x - 2) = (x + 3)(x - 1)$

$\rm :\longmapsto\: {x}^{2} + 5x - 2x - 10 = {x}^{2} + 3x - x - 3$

$\rm :\longmapsto\: 3x - 10 = 2x - 3$

$\rm :\longmapsto\: 3x -2x= 10 - 3$

$\bf\implies \:x = 7$

Additional Information :-

1. Pythagoras Theorem :-

This theorem states that : In a right-angled triangle, the square of the longest side is equal to sum of the squares of remaining sides.

2. Converse of Pythagoras Theorem :-

This theorem states that : If the square of the longest side is equal to sum of the squares of remaining two sides, angle opposite to longest side is right angle.

3. Area Ratio Theorem :-

This theorem states that :- The ratio of the area of two similar triangles is equal to the ratio of the squares of corresponding sides.

4. Basic Proportionality Theorem :-

If a line is drawn parallel to one side of a triangle, intersects the other two lines in distinct points, then the other two sides are divided in the same ratio.