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 $\overline{A C} ?$

,

$5$
$8$
$12$
$13$
$15$

Key Concepts

Geometry

Triangle

Pythagoras

Suggested Book | Source | Answer

Suggested Reading

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 $13$

Other useful links

Related

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 $\overline{A C} ?$

,

$5$
$8$
$12$
$13$
$15$

Key Concepts

Geometry

Triangle

Pythagoras

Suggested Book | Source | Answer

Suggested Reading

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 $13$

Other useful links

Related

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