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Olympiad workbook

Exam: Olympiad workbook
Subject: IOQM
Chapter: Properties of Triangles

Olympiad workbook · IOQM · Properties of Triangles

The ex-radii of a triangle are 10 (1)/(2), 12 and 14 . If the sides...

The ex-radii of a triangle are

10 \frac{1}{2}, 12

and 14 . If the sides of the triangle are the roots of the cubic

x^3-p x^2+q x-r=0

, where

p, q, r

are integers, find the integer nearest to

\sqrt{p+q+r}

.

Correct answer

58

Step-by-step solution

\because \quad r_1=\frac{\Delta}{S-a}=\frac{21}{2}, r_2=\frac{\Delta}{S-b}=12

and

r_3=\frac{\Delta}{S-c}=14

On solving above equations, we get

a=13, b=14, c=15

Let

f(x)=x^3-p x^2+q x-r

=(x-13)(x-14)(x-15)

f(-1)=-1-p-q-r

=(-14)(-15)(-16)

\therefore \quad p+q+r=14 \cdot 15 \cdot 16-1

\therefore \quad \sqrt{p+q+r}=\sqrt{3364} \approx 58

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