Question

Solve the question in the attachment!!

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Answer 1

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

Given that

ABCD is a rectangle such that AB = CD = 15 cm and AD = BC = 24 cm.

Further given that, PC = 5 cm

Let assume that QC = x cm

Now, In triangle QCP and triangle QBA

$\rm :\longmapsto\:\angle QCP = \angle QBA \: \: \{corresponding \: angles \}$

$\rm :\longmapsto\:\angle QPC = \angle QAB \: \: \{corresponding \: angles \}$

$\rm\implies \:\triangle QCP = \triangle QBA \: \: \: \{AA \: similarity \}$

$\rm\implies \:\dfrac{QC}{QB} = \dfrac{PC}{AB}$

$\rm :\longmapsto\:\dfrac{x}{x + 24} = \dfrac{5}{15}$

$\rm :\longmapsto\:\dfrac{x}{x + 24} = \dfrac{1}{3}$

$\rm :\longmapsto\:3x = x + 24$

$\rm :\longmapsto\:3x - x = 24$

$\rm :\longmapsto\:2x = 24$

$\rm :\longmapsto\:x = 12 \: cm$

$\rm\implies \:\boxed{\tt{ QC \: = \: 12 \: cm \: }}$

• Option (ii) is correct

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MORE TO KNOW

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 :-

This theorem states that: 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.

Answer 2

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

Given that

ABCD is a rectangle such that AB = CD = 15 cm and AD = BC = 24 cm.

Further given that, PC = 5 cm

Let assume that QC = x cm

Now, In triangle QCP and triangle QBA

$\rm :\longmapsto\:\angle QCP = \angle QBA \: \: \{corresponding \: angles \}$

$\rm :\longmapsto\:\angle QPC = \angle QAB \: \: \{corresponding \: angles \}$

$\rm\implies \:\triangle QCP = \triangle QBA \: \: \: \{AA \: similarity \}$

$\rm\implies \:\dfrac{QC}{QB} = \dfrac{PC}{AB}$

$\rm :\longmapsto\:\dfrac{x}{x + 24} = \dfrac{5}{15}$

$\rm :\longmapsto\:\dfrac{x}{x + 24} = \dfrac{1}{3}$

$\rm :\longmapsto\:3x = x + 24$

$\rm :\longmapsto\:3x - x = 24$

$\rm :\longmapsto\:2x = 24$

$\rm :\longmapsto\:x = 12 \: cm$

$\rm\implies \:\boxed{\tt{ QC \: = \: 12 \: cm \: }}$

Option (ii) is correct

▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬

MORE TO KNOW

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 :-

This theorem states that: 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.