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Olympiad workbook

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

Olympiad workbook · IOQM · Properties of Triangles

Let A B C be a right-angled triangle with B=90^ . Let the length of...

Let

A B C

be a right-angled triangle with

\angle B=90^{\circ}

. Let the length of the altitude

B D

be equal to 12 . What is the minimum possible length of

A C

, given that

A C

and the perimeter of triangle

A B C

are integers? Let A B C be a right-angled triangle with B=90^ . Let the length of the altitude B D be equal to 12 . What is the

Let A B C be a right-angled triangle with B=90^ . Let the length of the altitude B D be equal to 12 . What is the

Correct answer

25

Step-by-step solution

a c=12 b

Let

a+c=I \quad l \in

integer

a^2+c^2=R-24 b=b^2

Let A B C be a right-angled triangle with B=90^ . Let the length of the altitude B D be equal to 12 . What is the

\begin{aligned} & D=\geq 0 \\ & I^2-48 b \geq 0 \\ & b^2+24 I \geq 0 \quad b \geq 24 \\ & \text { Let } b=k-12 \quad k \geq 36 \\ & \begin{array}{l}R=k^2-144 \\ 144=(k-I)(k+I) \\ \text { For } k=37, I=35 \\ I=25 \text { is minimum value }\end{array}\end{aligned}

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