# How do I prove it is a diagonal?

1. Jul 27, 2016

### Zeroth

1. The problem statement, all variables and given/known data
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

2. Relevant equations

3. 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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2. Jul 27, 2016

### Staff: Mentor

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.

3. Jul 27, 2016

### Zeroth

Is this enough?

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4. Jul 27, 2016

### Staff: Mentor

Can't you type it in here? It's a) hard to read and b) skipped.

5. Jul 27, 2016

### Zeroth

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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6. Jul 27, 2016

### Staff: Mentor

What makes you think it is the diagonal? I guess you will have to determine the angles.

7. Jul 27, 2016

### ehild

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. )

View attachment 103906

Last edited: Jul 27, 2016
8. Jul 27, 2016

### Zeroth

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

9. Jul 27, 2016

### SammyS

Staff Emeritus
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. Jul 27, 2016

### Zeroth

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