Question

find x for the following quad

Answer 1

Step-by-step explanation:

answer in photo..

hope it helps !

Answer 2

$\large\underline{\bold{Given- }}$

A quadrilateral ABCD in which

• AB || CD

• OA = x + 5

• OB = x - 1

• OC = x - 3

• OD = x - 2

$\large\underline{\sf{To\:Find - }}$

• The value of 'x'

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

$\rm :\longmapsto\: \bf \: In \: \triangle \: BOA \: and \: \triangle \: DOC$

$\rm :\longmapsto\:\angle \: CDO \: = \: \angle \: ABO \: \: \{ \: alternate \: interior \: angles \}$

$\rm :\longmapsto\:\angle \: DOC \: = \: \angle \: BOA \: \: \{vertically \: opposite \: angles \}$

$\rm :\implies\:\triangle \: BOA \sim \triangle \: DOC \: \{AA \: similarity \}$

$\rm :\longmapsto\:\bf \: \therefore \: \dfrac{BO}{DO} = \dfrac{OA}{OC}$

On substituting the values of OA, OB, OC and OD, we get

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

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

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

$\rm :\longmapsto\: - 4x + 3 = 3x - 10$

$\rm :\longmapsto\: - 4x - 3x = - 10 - 3$

$\rm :\longmapsto\: - 7x = - 13$

$\bf\implies \:x = \dfrac{13}{7}$

Let verify the value of 'x'

So,

We find the dimensions of OC, OD, OA and OB.

$\rm :\longmapsto\:OC = x - 3 = \dfrac{13}{7} - 3 = - \: \dfrac{8}{3}$

$\sf \: which \: is \: not \: possible \: as \: side \: can \: never \: be \: negative.$

$\sf \: So, \: there \: is \: no \: value \: of \: x \: satisfying \: the \: condition.$

Additional Information :-

Important Theorems of Similar Triangles

1. Area Ratio Theorem :- If two triangles are similar, then the ratio of the area of both triangles is proportional to the square of the ratio of their corresponding sides.

2. Basic Proportionality Theorem :- If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points then the other two sides are divided in the same ratio.

3. Pythagoras Theorem :- Pythagoras theorem states that “In a right-angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two  sides“. The sides of this triangle have been named as Perpendicular, Base and Hypotenuse.

4. Converse of Pythagoras Theorem :- The converse of Pythagoras theorem states that “If the square of a side is equal to the sum of the square of the other two sides, then triangle must be right angle triangle”.