SATHEE: Properties of Triangles 1 Question 4

Properties of Triangles 1 Question 4

4. $A B C D$ is a trapezium such that $A B$ and $C D$ are parallel and $B C ⊥ C D$ , if $∠ A D B = θ, B C = p$ and $C D = q$ , then $A B$ is equal to

(2013 Main)

(a) $\frac{(p^{2} + q^{2}) \sin θ}{p \cos θ + q \sin θ}$

(b) $\frac{p^{2} + q^{2} \cos θ}{p \cos θ + q \sin θ}$

(d) $\frac{(p^{2} + q^{2}) \sin θ}{(p \cos θ + q \sin θ)^{2}}$

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Answer:

Correct Answer: 4. (a)

Solution:

Applying sine rule in $△ A B D$ ,

$\begin{aligned} \frac{A B}{\sin θ} & = \frac{\sqrt{p^{2} + q^{2}}}{\sin π - (θ + α)} \\ \Rightarrow \frac{A B}{\sin θ} & = \frac{\sqrt{p^{2} + q^{2}}}{\sin (θ + α)} \\ \Rightarrow A B & = \frac{\sqrt{p^{2} + q^{2}} \sin θ}{\sin θ \cos α + \cos θ \sin α} ∵ \cos α = \frac{q}{\sqrt{p^{2} + q^{2}}} \end{aligned}$

$= \frac{(p^{2} + q^{2}) \sin θ}{p \cos θ + q \sin θ}$

and $\sin α = \frac{p}{\sqrt{p^{2} + q^{2}}}$

Alternate Solution

Let $A B = x$

In $△ D A M, \tan (π - θ - α) = \frac{p}{x - q}$

$\Rightarrow \tan (θ + α) = \frac{p}{q - x}$

$\begin{aligned} \Rightarrow & q - x & = p \cot (θ + α) \\ \Rightarrow & x & = q - p \cot (θ + α) \end{aligned}$

$= q - p \frac{\cot θ \cot α - 1}{\cot α + \cot θ} ∵ \cot α = \frac{q}{p}$

$= q - p \frac{\frac{q}{p} \cot θ - 1}{\frac{q}{p} + \cot θ} = q - p \frac{q \cot θ - p}{q + p \cot θ}$

$= q - p \frac{q \cos θ - p \sin θ}{q \sin θ + p \cos θ}$

$\begin{array}{ll} \Rightarrow & x = \frac{q^{2} \sin θ + p q \cos θ - p q \cos θ + p^{2} \sin θ}{p \cos θ + q \sin θ} \\ \Rightarrow & A B = \frac{(p^{2} + q^{2}) \sin θ}{p \cos θ + q \sin θ} \end{array}$

Properties of Triangles 1 Question 4

4. $A B C D$ is a trapezium such that $A B$ and $C D$ are parallel and $B C ⊥ C D$ , if $∠ A D B = θ, B C = p$ and $C D = q$ , then $A B$ is equal to

Answer:

Alternate Solution

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Properties of Triangles 1 Question 4

4. ABCD is a trapezium such that AB and CD are parallel and BC⊥CD, if ∠ADB=θ,BC=p and CD=q, then AB is equal to

Answer:

Alternate Solution

Engineering Preparation

Medical Preparation

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Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

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NCERT Exercise Video Solution

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4. $A B C D$ is a trapezium such that $A B$ and $C D$ are parallel and $B C ⊥ C D$ , if $∠ A D B = θ, B C = p$ and $C D = q$ , then $A B$ is equal to