# Prove this is a right triangle in a sphere

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1. Sep 5, 2016

### Sho Kano

1. The problem statement, all variables and given/known data
Let P be a point on the sphere with center O, the origin, diameter AB, and radius r. Prove the triangle APB is a right triangle

2. Relevant equations
|AB|^2 = |AP|^2 + |PB|^2
|AB}^2 = 4r^2

3. The attempt at a solution
Not sure if showing the above equations are true is the correct way to do this problem

2. Sep 5, 2016

### LCKurtz

HInt: Try drawing a picture of triangle ABP with O at the center of AB. What do you know about OA, OB, and OP?

3. Sep 6, 2016

### Ray Vickson

The triangle ABP lies in a plane through the center of the sphere, so turns the three-dimensional problem into a 2-dimensional (planar) problem involving a circle of radius $r$.

4. Sep 6, 2016

### Sho Kano

Here's a very bad drawing (I forgot to insert the origin in, but lets just say it's there) (and let's say that the angle between BP and BA is x)

Is this sufficient?
|AB|^2 = |AP|^2 + |PB|^2
2r = d
|AP| = dsin(x)
|PB| = dcos(x)
|AB|^2 = d^2

d^2 = d^2 (cos(x)^2 + sin(x)^2)
d^2 = d^2

Last edited: Sep 6, 2016
5. Sep 6, 2016

### LCKurtz

You can't start with the Pythagorean theorem because you don't know that is a right triangle. That is what you are supposed to prove.

You aren't trying to prove $d^2 = d^2$.

6. Sep 6, 2016

### Sho Kano

But if it satisfies the pythagorean theorem then that must mean it's a right triangle right?

7. Sep 6, 2016

### LCKurtz

At the risk of repeating myself: You DON'T KNOW it satisfies the Pythagorean theorem because you don't know $x$ is a right angle.

8. Sep 7, 2016

### RUber

What methods and/or tools have you been learning recently that would be applicable to this problem?

Are you allowed to use the inscribed angle theorem?
https://en.wikipedia.org/wiki/Inscribed_angle

Or would using isosceles triangle rules be more appropriate?

9. Sep 7, 2016

### Sho Kano

I'm taking vector calculus and our teacher assigned this question for homework. I guess the methods we have been learning is the dot product and cross product and various vector properties.

10. Sep 7, 2016

### Sho Kano

I don't know where to start. Could you give me a hint? I know that the magnitude of OP where P is any Point in the circle is r.

11. Sep 7, 2016

### LCKurtz

This problem doesn't require any calculus at all. You have a triangle with isosceles sub-triangles. Label the angles. Almost anything you write down about the various angles should lead you to the solution.

12. Sep 7, 2016

### Sho Kano

By sub-triangles, you mean the triangle inscribed within the bigger one? How do I know that those triangles are isosceles?

13. Sep 7, 2016

### LCKurtz

Here is what you wrote in post #10:

"I don't know where to start. Could you give me a hint? I know that the magnitude of OP where P is any Point in the circle is r."

14. Sep 7, 2016

### Sho Kano

Don't see how that automatically makes a triangle inscribed within triangle APB an isosceles. I don't see it.

15. Sep 7, 2016

### LCKurtz

Draw the picture I suggested in post #2, including the point O. The smaller triangles all have the point O as a vertex. Use the ideas you have been given. I can't do any more for you without working the problem for you.

16. Sep 8, 2016

### RUber

Look at it as two triangles, AOP and POB. Let one angle at O (say angle AOP) have measure z, then what is the measure of angle POB?
Notice that AO is a radius, and OP is a radius. This should give you a clue about what LCKurtz was talking about.
Remember that isosceles triangles have equal base angles and the sum of interior angles sum to 180.
This should be enough info for you to do the proof geometrically.

Using vectors, if you use the unit circle with diameter points at (-1,0) and (1,0) for simplicity, then for any point where $\theta \neq 0, \pi$, your point P is at $\cos \theta \hat x + \sin \theta \hat y$. You can use the dot product of the vectors describing the legs to show that the vectors are orthogonal.
You could generalize to a non-unit circle by using diameter points at (-r,0) and (r,0) and point P at $r\cos \theta \hat x + r\sin \theta \hat y$.

17. Sep 8, 2016

### Sho Kano

Think I got it, from the figure on post #4
$PA\cdot PB=0\\ PA=OA-OP\\ PB=OB-OP\\ (OA-OP)\cdot (OB-OP)=0\\ OA\cdot OB-OA\cdot OP-OP\cdot OB+OP\cdot OP=0\\ OA=-OB,\quad OB\cdot OB={ r }^{ 2 },\quad OP\cdot OP={ r }^{ 2 }\\ -{ r }^{ 2 }-OA\cdot OP-OP\cdot OB+{ r }^{ 2 }=0\\ OP\cdot (-OA-OB)=0\\ OP\cdot (-OA-(-OA))=0\\ OP\cdot (O)=0$

18. Sep 8, 2016

### RUber

It looks like you are starting with what you want to prove. Other than that, I think your expansion and logic look good.

19. Sep 8, 2016

### ehild

If you do what LCKurtz (Post #15) suggested and draw the line OP, it cuts the triangle into two isosceles ones. The angle APB is β+γ. What is its value?

20. Sep 8, 2016

### Sho Kano

I see why they are isosceles now, then
2γ = α
2γ + 2β = 180
γ + β = 90
This way is much faster haha