# Circle theorem-Chord

1. Nov 20, 2013

One rule of circle theorems is,a line drawn from center to the mid-point of a chord cuts the cord at 90°.
What's the proof?
It's true that two radius and a chord creates an isosceles triangle.
So,

How can I prove that in an isosceles triangle,a line drawn from the vertex angle to the mid-point of the base side cuts the side at 90°?

#### Attached Files:

• ###### Chord.gif
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Last edited: Nov 20, 2013
2. Nov 20, 2013

### Staff: Mentor

You don't (yet) know that those two angles are 90 degrees.... But what do you know about them?

3. Nov 20, 2013

I'm not talking about the base anlgles.I am talking about the 90° angle you see in the image.
let vertex angle be y
$180-(\frac{y}{2}+x)=90$
$2x+y=180$

I can't seem to solve y and x.I don't know whether this is a proof or not.

4. Nov 20, 2013

### Staff: Mentor

I'm sorry, I wasn't clear. Those two 90-degree angles are the two angles that I mean, and my question still stands: What do you know about them?

5. Nov 20, 2013

hahahah.

I don't know what do you mean.
In a right angle triangle, $c^2=a^2+b^2$

c and b of both triangles is same.
so if it is two right angle triangles,a should be same
As the line was drawn to the midpoint,a is same in both triangles.So it is a right angle triangle.
Is this enough for a proof?

Last edited: Nov 20, 2013
6. Nov 20, 2013

### Staff: Mentor

No, because you've worked in the assumption both are right triangles, which is what you're trying to prove. The $c^2=a^2+b^2$ relationship isn't helping any because you don't know the values for all three to prove that you do have a right triangle....

But there's a reason I keep asking what you know about the two angles, not just one of them... What is their sum?

7. Nov 20, 2013

180°.Where any straight two lines intercept,any one side of the line has 180°.
but why should this help?If we have a 40° and a 50° angle,we would still get 180 as a sum.

8. Nov 20, 2013

### HallsofIvy

Your midpoint line divides the large triangle into two smaller congruent triangles. They are congruent by "SSS": the two radii of the circle, the two bases, and the single ray in both triangles. From that follows that other "corresponding parts" are congruent.

9. Nov 21, 2013

I know it's congruent.but how do I know that the angle is 90°?
I know that if vertex angle is y,the upper angle of both triangles have to be y/2

10. Nov 21, 2013

### Tanya Sharma

Look at the attached figure.

Since ΔABD is congruent to ΔACD ,

#### Attached Files:

• ###### Chord.gif
File size:
4.4 KB
Views:
74
11. Nov 21, 2013