How do I prove it is a diagonal?

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To prove that point M is on diagonal AC of square ABCD, the Pythagorean theorem can be applied using the distances MA, MB, and MC. The discussion highlights the need to calculate angles and side lengths to confirm M's position. Participants suggest using the Law of Sines and Pythagorean relationships to derive necessary values. A sketch is mentioned as a helpful tool for visualizing the problem, but clarity in the calculations is emphasized. Understanding these geometric principles will ultimately lead to finding the area of the square.
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Homework Statement


Inside of a square ABCD exists a point M such that |MA|=7 cm, |MB|=13 cm and |MC| = 17 cm. Calculate the area of the square

Homework Equations

The Attempt at a Solution


After a lot of formula searching I went and looked at the solution, there they proved that the point M i on the diagonal AC using the coordinate system and from there on it is easy to find the area. I need help in understanding the proof. They mentioned it is possible to do it using Pythagorean therorem but they didn't show how to do it and I don't see it. You can see my sketch in the picture. I know I am supposed to show my work and I have nothing to show, but can you at least give me some guidelines please?
 

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Have you tried the law of sines to determine all relevant angles or some Pythagoras by drawing the heights of your triangles?
However, the first seems to be easier.
 
Is this enough?
 

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Can't you type it in here? It's a) hard to read and b) skipped.
 
X/sin(γ) = 13/sin(α)

X/sin(δ)= 13/sin(β)

That is only possible if γ=δ and α=β

I hope the pic is understandable.

EDIT: Also.. I'm on the phone so it's a bit hard to write
 

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What makes you think it is the diagonal? I guess you will have to determine the angles.
 
Zeroth said:
X/sin(γ) = 13/sin(α)

X/sin(δ)= 13/sin(β)

That is only possible if γ=δ and α=β

I hope the pic is understandable.

EDIT: Also.. I'm on the phone so it's a bit hard to write
It does not follow from your equations that γ=δ and α=β.

Look at the picture , apply Pythagoras' theorem and find x, y. (The red lines go through the point M and are parallel to the sides of the square. )
upload_2016-7-27_21-5-7.png
https://www.physicsforums.com/attachments/103906
 
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I realized the problem with the angles later. With that picture I had no problem doing it. In the solutions was essentially the same thing but the coordinate system made it very confusing. Thank you very much
 
Zeroth said:

Homework Statement


Inside of a square ABCD exists a point M such that |MA|=7 cm, |MB|=13 cm and |MC| = 17 cm. Calculate the area of the square

Homework Equations

The Attempt at a Solution


After a lot of formula searching I went and looked at the solution, there they proved that the point M is on the diagonal AC using the coordinate system and from there on it is easy to find the area. I need help in understanding the proof. They mentioned it is possible to do it using Pythagorean therorem but they didn't show how to do it and I don't see it. You can see my sketch in the picture. I know I am supposed to show my work and I have nothing to show, but can you at least give me some guidelines please?
If you take their word for it, that "the point M is on the diagonal AC", then the Law of cosines can be used to find the length of a side.

However, it's not too difficult to use the Pythagorean Theorem to show that M is on diagonal AC. From that finding the length of a side is easy.

Place point, P, along segment MC a distance 5 cm from M . What can you determine about triangle MPB ?
 
  • #10
SammyS said:
However, it's not too difficult to use the Pythagorean Theorem to show that M is on diagonal AC. From that finding the length of a side is easy.

Place point, P, along segment MC a distance 5 cm from M . What can you determine about triangle MPB ?

I will be completely honest with you. I ddon't see eye to eye with geometry. I can't determine anything.
 
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