Math Is Fun Forum

Daniel123

Member

Registered: 2007-05-23

Posts: 663

Prove isosceles and right-angled

In a triangle, the largest side has length 2cm and one of the other sides has length √2 cm. Given that the area of the triangle is 1cm², show that the triangle is right-angled and isosceles.

I'm confused... the only way I can seem to do it is by using the fact that it is right-angled and isosceles to show that it is right-angled and isosceles.. but I don't think that proves anything.

Does anyone mind showing me how to do it?

Thanks.

bossk171

Member

Registered: 2007-07-16

Posts: 305

Re: Prove isosceles and right-angled

side a = √2
side c = 2

Use the law of cosines to get b in terms of the cosine of the angle opposite side c:

Use the quadratic formula:

From that we can deduce that if gamma is 90° then b is √2.

Now we can use a modified version of Heron's formula:

seeing how Area (A) is 1, we can say:

Subbing our original values for a and b:

so, b is ±√2 or ±√10

Now, we've seen already that b is √2 when gamma is 90° But what of the other three values? I don't know how to say it technically, but it has something to do with the fact that they're negative angles (note √10 > pi).

Last edited by bossk171 (2007-10-23 06:44:28)

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There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction.