Question

In a 45°-45°- 90° triangle, if the legs each have a length of 4, then what is the length of the hypotenuse? A. the square root of2
B. 4the square root of2
C. 4
D. 4the square root of3

Answer 1

Do you know the ratio of the side-lengths to the hypotenuse?

Otherwise we could also use Pythagorean theorem if we had forgotten.

Answer 2

I just needed this one explained.. I did not understand how to do special right triangles.

Answer 3

On one hand, we have some specific lengths to memorize. For any 45-45-90 triangle, there is a specific proportion relating the length of the legs to the length of the hypotenuse.

It is normally a task of simply memorizing those ratios to make this very painless. But it all rolls back to the Pythagorean theorem for proof methods.

Say we had a leg side length of s and we call the hypotenuse h.
Then Pythagorean thm tells us that: s^2 + s^2 = h^2.

We simplify by adding the like terms: 2s^2 = h^2
And then the square root of both sides gives us: sqrt(2) s = h.
This general statement always holds true given a leg side-length s or a hypotenuse length h, for all 45-45-90 triangles...
makes sense so far?

Answer 4

The reason we really have the special triangles, though, is for a simplification reason.
I showed that sqrt(2) * s = h for all 45-45-90 triangles. If we simply know this to be true, we erase some work in the process.

Instead of writing out the Pythagorean thm, we just say "We know we have a 45-45-90 triangle. And the side length is 4. Now we just plug it in: h = 4 * sqrt(2) "

Answer 5

Just like if we had the hypotenuse already, say h = 45 sqrt(2), the equation also holds:
45 sqrt(2) = s * sqrt(2)
45 = s

Answer 6

So in other words, without knowing the hypotenuse, it would be 4 sqrt(2)?

Answer 7

Yes. In general, given a leg side-length of the 45-45-90 triangle, we simply multiply by sqrt(2) to obtain our hypotenuse.
And given the hypotenuse, the reverse direction of dividing by sqrt(2) gets us back to side length.

Answer 8

So final answer would be sqrt(2)?

Answer 9

We already had the hypotenuse as 4sqrt(2). What did you do to get sqrt(2) only?

Answer 10

Idk.. sorry lol

Answer 11

No problem. :p

Answer 12

Could you show me how to find X on a triangle from a special right 45, and 30? Ive got a few pics

Answer 13

For the first one.
9 is in the same place as which part (s, s sqrt(3), or 2s) in the 30-60-90 triangle I drew? And then for x?

Answer 14

The s.

Answer 15

Hang on...

Answer 16

And the x is the hypotenuse, SQRT(2)

Answer 17

you scribbling that out confuses me..lol

Answer 18

Okay, let me redraw that.