SATHEE: Properties of Triangles 3 Question 9

Properties of Triangles 3 Question 9

9. In a $△ P Q R$ , let $∠ P Q R = 30^{\circ}$ and the sides $P Q$ and $Q R$ have lengths $10 \sqrt{3}$ and 10 , respectively. Then, which of the following statement(s) is (are) TRUE? (2018 Adv)

(a) $∠ Q P R = 45^{\circ}$

(b) The area of the $△ P Q R$ is $25 \sqrt{3}$ and $∠ Q R P = 120^{\circ}$

(d) The area of the circumcircle of the $△ P Q R$ is $100 π$

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Answer:

Correct Answer: 9. $(b, c, d)$

Solution:

We have, In $△ P Q R$

$\begin{aligned} ∠ P Q R & = 30^{\circ} \\ P Q & = 10 \sqrt{3} \\ Q R & = 10 \end{aligned}$

By cosine rule

$\begin{aligned} \cos 30^{\circ} & = \frac{P Q^{2} + Q R^{2} - P R^{2}}{2 P Q \cdot Q R} \\ \Rightarrow & \frac{\sqrt{3}}{2} & = \frac{300 + 100 - P R^{2}}{200 \sqrt{3}} \\ \Rightarrow 300 & = 300 + 100 - P R^{2} \\ \Rightarrow P R & = 10 \\ Since, PR = Q R = 10 \\ ∴ & ∠ Q P R & = 30^{\circ} and ∠ Q R P = 120^{\circ} \\ Area of △ P Q R & = \frac{1}{2} P Q \cdot Q R \cdot \sin 30^{\circ} \\ = \frac{1}{2} \times 10 \sqrt{3} \times 10 \times \frac{1}{2} = 25 \sqrt{3} \end{aligned}$

Radius of incircle of

$Δ P Q R = \frac{Area of △ P Q R}{Semi - perimetre of Δ P Q R}$

$\begin{array}{cc} i.e. & r = \frac{Δ}{s} = \frac{25 \sqrt{3}}{\frac{10 \sqrt{3} + 10 + 10}{2}} = \frac{25 \sqrt{3}}{5 (\sqrt{3} + 2)} \\ \Rightarrow r = 5 \sqrt{3} (2 - \sqrt{3}) \\ = 10 \sqrt{3} - 15 \end{array}$

and radius of circumcircle $(R) = \frac{a b c}{4 Δ} = \frac{10 \sqrt{3} \times 10 \times 10}{4 \times 25 \sqrt{3}} = 10$

$∴$ Area of circumcircle of

$Δ P Q R = π R^{2} = 100 π$

Hence, option (b), (c) and (d) are correct answer.

Properties of Triangles 3 Question 9

9. In a $△ P Q R$ , let $∠ P Q R = 30^{\circ}$ and the sides $P Q$ and $Q R$ have lengths $10 \sqrt{3}$ and 10 , respectively. Then, which of the following statement(s) is (are) TRUE? (2018 Adv)

Answer:

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Properties of Triangles 3 Question 9

9. In a △PQR, let ∠PQR=30∘ and the sides PQ and QR have lengths 103 and 10 , respectively. Then, which of the following statement(s) is (are) TRUE? (2018 Adv)

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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9. In a $△ P Q R$ , let $∠ P Q R = 30^{\circ}$ and the sides $P Q$ and $Q R$ have lengths $10 \sqrt{3}$ and 10 , respectively. Then, which of the following statement(s) is (are) TRUE? (2018 Adv)