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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

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 ACwhich 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

Subscribe to Cheenta at Youtube


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 ACwhich 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

Subscribe to Cheenta at Youtube


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