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# Surds in geometry question watch

1. An isosceles right-angled triangle has its two shorter sides of length a. Write down an expression for its perimeter in terms of a.

A length of rope 10 meters long is to be pegged out to form an isosceles right-angled triangle. Find in as simple a form as possible, exact expressions for the lengths of the sides.

I'm totally lost at this question. I've got the answers in the back of the book but no idea of how to get there. Can somebody explain the process fully? Thanks
2. "isosceles right-angled triangle"...so you know exactly what the angle are.
"two shorter sides of length a"...so you know which sides they must be, can you work out the other side using pythagoras?
3. (Original post by Bleak Lemming)
An isosceles right-angled triangle has its two shorter sides of length a. Write down an expression for its perimeter in terms of a.

A length of rope 10 meters long is to be pegged out to form an isosceles right-angled triangle. Find in as simple a form as possible, exact expressions for the lengths of the sides.

I'm totally lost at this question. I've got the answers in the back of the book but no idea of how to get there. Can somebody explain the process fully? Thanks
For the first part, you need to work out the length of the hypotenuse. The two shorter sides are both length a, so by using Pythagoras' theorem the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the two shorter sides. This gives C² = a² + a² where C is the length of the hypotenuse. Square rooting both sides will give C = sqrt(2a²) = sqrt(2)*a. Adding all 3 sides together will give the perimeter.

For the second part, you just need equate this to 10 and then solve for a, and substitute this in to the expression for the hypotenuse.
4. (Original post by Phil_Waite)
For the first part, you need to work out the length of the hypotenuse. The two shorter sides are both length a, so by using Pythagoras' theorem the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the two shorter sides. This gives C² = a² + a² where C is the length of the hypotenuse. Square rooting both sides will give C = sqrt(2a²) = sqrt(2)*a. Adding all 3 sides together will give the perimeter.

For the second part, you just need equate this to 10 and then solve for a, and substitute this in to the expression for the hypotenuse.
Yea the first part is

I just don't get where came from...
5. (Original post by Bleak Lemming)
Yea the first part is

I just don't get where came from...
It's because the two shorter sides cannot be the hypoteneuse (otherwise the triangle does not exist) and thus the length of the hypoteneuse (say, C) is given by (as it is a right-angled triangle):

Then clearly the comes from the other two sides.
6. (Original post by marcusmerehay)
It's because the two shorter sides cannot be the hypoteneuse (otherwise the triangle does not exist) and thus the length of the hypoteneuse (say, C) is given by (as it is a right-angled triangle):

Got it, thanks :]
7. You should get the perimeter as a + a + sqrt(a^2) = 2a + sqrt(2)a = a( 2 + sqrt(2)).
8. 1) To Find the hypotenuse, since its a right angled triangle and isoceles, 2 sides will be of length a, and we find the hypotenuse.
Hypotenuse

Perimeter

2) Since they say that the perimeter is now 10 metres,

Therefore you know the two sides, and the hypotenuse is then
Hypotenuse
9. (Original post by Bleak Lemming)
Yea the first part is

I just don't get where came from...
Do you understand the part where the square of the hypotenuse is equal to the sum of the squares of both sides? C² = a²+a² where C is the length of the hypotenuse. If you take the square root of both sides, you get:

Adding this to the length of the other two sides:

Edit - Too late
10. If anybody also knows LaTex properly can you please help me to mend the 10/2+sqrt2 part
11. Thats what I wrote. I don't seem to understand why it looks like that

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