Can someone tell me why a triangle inscribed in a semicircle is always a right triangle?

I am in Trig currently and someone told me I had to use the laws of cosine. Which I am familiar with, I just don't know how to prove it.

Thank you

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- Aug 1st 2008, 10:44 AMzodiacbravetriangle inscribed in a semicircle
Can someone tell me why a triangle inscribed in a semicircle is always a right triangle?

I am in Trig currently and someone told me I had to use the laws of cosine. Which I am familiar with, I just don't know how to prove it.

Thank you - Aug 1st 2008, 11:14 AMearboth
This is a special case of Inscribed and Central Angles in a Circle from Interactive Mathematics Miscellany and Puzzles

When you read the proof keep in mind that your central angle is 180°. - Aug 1st 2008, 04:49 PMticbol
If it should be proven by the Law of Cosines, then,

let yhe two legs of the inscribed triangle be x and y

the hypotenuse is diameter d.

we assume the angle between x and r is 90 degrees.....this angle is opposite d.

By Law of Cosines,

d^2 = x^2 +y^2 -2(x)(y)cos(90deg)

d^2 = x^2 +y^2 -2xy(0)

d^2 = x^2 +y^2 -------------this is the Pythagorean Theorem, applicable to right triangles only. So the inscribed triangle is really a right triangle, whatever the lengths of x and y are. - Aug 1st 2008, 06:26 PMQuick

It sometimes helps to see a dynamic diagram.

There should be a java applet to the right of this text (it might take a second to load)

You can move the red dots around.

Notice that angle C is always a right angle.

On a side note to others: How many knew that you could type in a java applet? I used to use it all the time, but I haven't been here in like a year. - Aug 1st 2008, 07:52 PMSoroban
Hello, zodiacbrave!

Well, here's one way . . . my way.

Quote:

Why is a triangle inscribed in a semicircle is always a right triangle?

Draw chords and

LetCode:`* * *`

* * C

* o

* r * *

*

* θ * θ' *

A o - - - - o - - - - o B

* r O r *

* *

* *

* *

* * *

Use the Law of Cosines in

. . We have: . .[1]

Use the Law of Cosines in

. . We have: . .[2]

Since

. .

Substitute into [2]: .

. . and we have: . .[3]

Add [1] and [3]: .

. . . . . Pythagorus!

Therefore, is a right triangle with