Pre-RMO Question!

Geometry Level 4

Let $AD$ be an altitude in a right triangle $ABC$ with $\angle A$ $= 90^\circ$ and $D$ on $BC.$ Suppose that the radii of the incircles of the triangles $ABD$ and $ACD$ are $33$ and $56$ respectively. Let $r$ be the radius of the incircle of triangle $ABC.$ Find the value of $3(r + 7).$

The answer is 216.

1 solution

Maria Kozlowska
Sep 11, 2016

Triangles $ABC, ABD, ACD$ are similar. $\frac{AB}{AC}=\frac{33}{56}$ . $\frac{BC}{AC}=\frac{\sqrt{33^2 + 56^2}}{56}=\frac{65}{56} \Rightarrow r=65$ .

$3(65+7)=\boxed{216}$ .

edit your solution.It is 3(65+7) and not 3(4+7).Rather a very good solution and observation...+1

Ayush G Rai - 4 years, 9 months ago

Can u explain this solution please How is AB/BC=33/56

A Former Brilliant Member - 4 years, 8 months ago

Because triangles are similar their corresponding sides lengths are in the same proportion as their in-radii.

Maria Kozlowska - 4 years, 8 months ago

@Maria Kozlowska – can u prove this

A Former Brilliant Member - 4 years, 8 months ago

@A Former Brilliant Member – It follows from the formula for the in-radius: $r=\frac{\triangle}{s}$ where s is semiperimeter. If we use the Heron's formula for the area of the triangle we can deduce that the in-radii of two similar triangles are in a same proportion as their corresponding sides.

Maria Kozlowska - 4 years, 8 months ago

@Maria Kozlowska – thanks! I didn't know this property

A Former Brilliant Member - 4 years, 8 months ago

@A Former Brilliant Member – It is intuitive. In similar shapes all corresponding lengths have to be in the same proportion.

Maria Kozlowska - 4 years, 8 months ago

Did the exact same

Aditya Kumar - 4 years, 9 months ago

How is AB/AC= 33/56 Please explain your solution it is not clear

A Former Brilliant Member - 4 years, 8 months ago

Pre-RMO Question!

The answer is 216.

1 solution

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