SATHEE: Properties of Triangles 3 Question 3

Properties of Triangles 3 Question 3

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

(2014 Adv)

(a) $\frac{3 y}{2 x (x + c)}$

(b) $\frac{3 y}{2 c (x + c)}$

(d) $\frac{3 y}{4 c (x + c)}$

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Answer:

Correct Answer: 3. (b)

Solution:

PLAN (i) $\cos C = \frac{a^{2} + b^{2} - c^{2}}{2 a b}$ (ii) $R = \frac{a b c}{4 Δ}, r = \frac{Δ}{s}$

where, $R, r, Δ$ denote the circumradius, inradius and area of triangle, respectively.

Let the sides of triangle be $a, b$ and $c$ .

$\begin{aligned} Given, \\ x = a + b \\ y = a b \\ x^{2} - c^{2} = y \\ \Rightarrow (a + b)^{2} - c^{2} = y \\ \Rightarrow a^{2} + b^{2} + 2 a b - c^{2} = a b \\ \Rightarrow a^{2} + b^{2} - c^{2} = - a b \\ \Rightarrow \frac{a^{2} + b^{2} - c^{2}}{2 a b} = - \frac{1}{2} = \cos 120^{\circ} \\ \Rightarrow ∠ C = \frac{2 π}{3} \\ ∵ R = \frac{a b c}{4 Δ}, r = \frac{Δ}{s} \Rightarrow \frac{r}{R} = \frac{4 Δ^{2}}{s (a b c)} \\ = \frac{4 \frac{1}{2} a b \sin \frac{2 π}{3}^{2}}{\frac{x + c}{2} \cdot y \cdot c} \\ ∴ \frac{r}{R} = \frac{3 y}{2 c (x + c)} \end{aligned}$

Properties of Triangles 3 Question 3

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

Answer:

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Properties of Triangles 3 Question 3

3. In a triangle, the sum of two sides is x and the product of the same two sides is y. If x2−c2=y, where c is the third side of the triangle, then the ratio of the inradius to 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 two sides is $x$ and the product of the same two sides is $y$ . If $x^{2} - c^{2} = y$ , where $c$ is the third side of the triangle, then the ratio of the inradius to the circumradius of the triangle is