SATHEE: Properties of Triangles 1 Question 3

Properties of Triangles 1 Question 3

3. In a triangle, the sum of lengths of two sides is $x$ and the product of the lengths of the same two sides is $y$ . If $x^{2} - c^{2} = y$ , where $c$ is the length of the third side of the triangle, then the circumradius of the triangle is

(2019 Main, 11 Jan I)

(a) $\frac{c}{3}$

(b) $\frac{c}{\sqrt{3}}$

(d) $\frac{y}{\sqrt{3}}$

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Answer:

Correct Answer: 3. (b)

Solution:

We know that, $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2 R$ and given that, $a + b = x, a b = y$ and $x^{2} - c^{2} = y$

$\begin{array}{ll} ∴ & (a + b)^{2} - c^{2} = a b \\ \Rightarrow & a^{2} + b^{2} - c^{2} = - 2 a b + a b \\ \Rightarrow & a^{2} + b^{2} - c^{2} = - a b \\ \Rightarrow & \frac{a^{2} + b^{2} - c^{2}}{2 a b} = \frac{- a b}{2 a b} = - \frac{1}{2} \end{array}$

$∴ \cos C = - \frac{1}{2} \Rightarrow C = 120^{\circ}$

[using cosine rule, $\cos C = \frac{a^{2} + b^{2} - c^{2}}{2 a b}$ ]

Now, $\frac{c}{\sin C} = 2 R$

$\begin{array}{ll} \Rightarrow & R = \frac{1}{2} \frac{c}{\sin (120^{\circ})} = \frac{c}{2} \frac{2}{\sqrt{3}} \\ ∴ & R = \frac{c}{\sqrt{3}} \end{array}$

Properties of Triangles 1 Question 3

3. In a triangle, the sum of lengths of two sides is $x$ and the product of the lengths of the same two sides is $y$ . If $x^{2} - c^{2} = y$ , where $c$ is the length of the third side of the triangle, then the circumradius of the triangle is

Answer:

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Properties of Triangles 1 Question 3

3. In a triangle, the sum of lengths of two sides is x and the product of the lengths of the same two sides is y. If x2−c2=y, where c is the length of the third side of the triangle, then the circumradius of the triangle is

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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3. In a triangle, the sum of lengths of two sides is $x$ and the product of the lengths of the same two sides is $y$ . If $x^{2} - c^{2} = y$ , where $c$ is the length of the third side of the triangle, then the circumradius of the triangle is