SATHEE: Properties of Triangles 3 Question 12

Properties of Triangles 3 Question 12

12. In a $△ A B C, a : b : c = 4 : 5 : 6$ . The ratio of radius of the circumcircle to that of the incircle is… .

(1996, 1M)

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Answer:

Correct Answer: 12. $\frac{16}{7}$

Solution:

We have, $R = \frac{a b c}{4 Δ}$ and $r = \frac{Δ}{s}$

$\begin{aligned} \frac{R}{r} & = \frac{a b c}{4 Δ} \cdot \frac{s}{Δ} = \frac{a b c \cdot s}{4 Δ^{2}} \\ = \frac{a b c}{4 (s - a) (s - b) (s - c)} \end{aligned}$

But $a : b : c = 4 : 5 : 6$

[given]

$\begin{aligned} \Rightarrow \frac{a}{4} = \frac{b}{5} = \frac{c}{6} = k \\ \Rightarrow \\ Now, s = \frac{1}{2} (a + b + c) = \frac{1}{2} (4 k + 5 k + 6 x) = \frac{15 k}{2} \\ ∴ \frac{R}{r} = \frac{(4 k) (5 k) (6 k)}{4 \frac{15 k}{2} - 4 k \frac{15 k}{2} - 5 k \frac{15 k}{2} - 6 k} \\ = \frac{30 k^{3}}{k^{3} \frac{15 - 8}{2}} \frac{\frac{15 - 10}{2}}{\frac{15 - 12}{2}} = \frac{30 \cdot 8}{7 \cdot 5 \cdot 3} = \frac{16}{7} \end{aligned}$

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Properties of Triangles 3 Question 12

12. In a △ABC,a:b:c=4:5:6. The ratio of radius of the circumcircle to that of the incircle is… .

Answer:

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12. In a $△ A B C, a : b : c = 4 : 5 : 6$ . The ratio of radius of the circumcircle to that of the incircle is… .