Algebraic Proof for x in a Right Triangle: PS, SR, and RP on a Straight Line

In summary, the conversation discusses using algebra to show that in a diagram of a right triangle, the length of the side QR can be expressed as x = (b^2 + c^2 - a^2) / 2c. This can be achieved by using Pythagoras' theorem on both similar triangles formed by drawing a perpendicular line from the right angle to the hypotenuse. By simplifying the equation, the desired result can be obtained.
  • #1
Natasha1
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Homework Statement


In a diagram, P, Q and R lie on a straight line and angle SQR is a right angle. The lengths PS, SR and RP are a, b and c cm respectively and QR is x cm.

Use algebra to show that x = (b^2 + c^2 - a^2) / 2c

Homework Equations

and attempt at a solution[/B]

Do I need to use Pythagoras for this question? I need a prompt, a little help please...
 

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  • #2
Natasha1 said:

Homework Statement


In a diagram, P, Q and R lie on a straight line and angle SQR is a right angle. The lengths PS, SR and RP are a, b and c cm respectively and QR is x cm.

Use algebra to show that x = (b^2 + c^2 - a^2) / 2c

Homework Equations

and attempt at a solution[/B]

Do I need to use Pythagoras for this question? I need a prompt, a little help please...
Beats me. Do you have a diagram to refer to? If not, you should make one, based on the description of the points.

Since there is a right angle involved, I would keep Pythagoras handy, nevertheless.
 
  • #3
I have attached a picture of the triangle in question... Hope it worked...
 
  • #4
Drawing a line from the right angle perpendicular to the hypotenuse divides the original right angle into two triangles both similar to the original triangle.
 
  • #5
Natasha1 said:
Do I need to use Pythagoras for this question? I need a prompt, a little help please...

Yes, you can solve the problem by using Pythagoras' Theorem for both the yellow and blue triangles.

upload_2016-3-22_6-32-52.png
 
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Likes Natasha1
  • #6
Got it!
After some working I got...

a^2 - c^2 +2cx -x^2 = b^2 - x^2
which simplifies to:
x = (b^2 + c^2 - a^2) / 2c

Thanks ehild I equated the m
 
  • #7
well done!
 

What is the Pythagorean Theorem?

The Pythagorean Theorem is a mathematical principle that states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

Who was Pythagoras?

Pythagoras was an ancient Greek mathematician and philosopher who is best known for his work in geometry and is credited with discovering the Pythagorean Theorem.

How is algebra related to the Pythagorean Theorem?

Algebra is used to represent the Pythagorean Theorem in a formula, where a and b are the lengths of the two legs of a right triangle and c is the length of the hypotenuse: a² + b² = c². Algebra can also be used to solve for any of the three sides of a right triangle given the other two lengths.

What are some real-world applications of the Pythagorean Theorem?

The Pythagorean Theorem has many practical applications in fields such as architecture, engineering, and physics. It can be used to calculate distances, heights, and areas in real-world scenarios, such as determining the length of a ladder needed to reach a certain height on a building.

Is the Pythagorean Theorem limited to right triangles?

Yes, the Pythagorean Theorem only applies to right triangles, which are triangles with one angle that measures 90 degrees. It does not work for other types of triangles, such as equilateral or scalene triangles.

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