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# Right-angled Triangle | AMC 10A, 2018 | Problem No 16

Try this beautiful Problem on Geometry based on Right-angled triangle from AMC 10 A, 2018. You may use sequential hints to solve the problem.

## Right-angled triangle - AMC-10A, 2018- Problem 16

Right triangle $A B C$ has leg lengths $A B=20$ and $B C=21$. Including $\overline{A B}$ and $\overline{B C}$, how many line segments with integer length can be drawn from vertex $B$ to a point on hypotenuse $\overline{A C} ?$

,

• $5$
• $8$
• $12$
• $13$
• $15$

Geometry

Triangle

Pythagoras

## Suggested Book | Source | Answer

Pre College Mathematics

#### Source of the problem

AMC-10A, 2018 Problem-16

#### Check the answer here, but try the problem first

$13$

## Try with Hints

#### First Hint

Given that $\triangle ABC$ is a Right-angle triangle and $AB=20$ and $BC=21$. we have to find out how many line segments with integer length can be drawn from vertex $B$ to a point on hypotenuse $\overline{AC}$?

Let $P$ be the foot of the altitude from $B$ to $AC$. therefore $BP$ is the shortest legth . $B P=\frac{20 \cdot 21}{29}$ which is between $14$ and $15$.

Now can you finish the problem?

#### Second Hint

let us assume a line segment $BY$ with $Y$ on $AC$which is starts from $A$ to $P$ . So if we move this line segment the length will be decreases and the values will be look like as $20,.....,15$. similarly if we moving this line segment $Y$ from $P$ to $C$ hits all the integer values from $15, 16, \dots, 21$.

Now Can you finish the Problem?

#### Third Hint

Therefore numbers of total line segments will be $13$

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