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
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
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°.
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.
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.
Hello, zodiacbrave!
Well, here's one way . . . my way.
We have a circle with center , diameter , and radiiWhy 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