# What is the length of the legs of the triangle?

In a 30-60-90 degree triangle, my hypotenuse is 1 unit. What is the length of the two legs?

I think I should use 'a squared plus b squared equals c squared' but Idk?

### 6 Answers

- 1 month agoFavourite answer
Looking at the problem above, clearly an isoceles right triangles means that two legs are equal, since the hypothenuse is always the longest side of the right triangle.

Obviously, the isoceles right triangle is a 45-45-90.

Using the Pythagorean Theorem, you are to reference the legs as being equal.

a^2 + b^2 = c^2

if the hypothenuse is c and the hypthenuse is 1

Then, you can rewrite the expression as a^2 + b^2 = 1, since 1^2 = 1

since the legs are both the same, you can rewrite the expression with the same variable for each side of the right triangle.

Let's say I want to use x for a leg. Since they are both the same, each leg will be represented by the x variable for good measure

So, I will replace x for both a and b in the equation so that the equation, a^2 + b^2 = 1, becomes x^2 + x^2 = 1.

because the exponent of x are the same, we can add the coefficient value of 1 found in x^2 and the other x^2 to give a simplified value of 2x^2 = 1

Perform inverse order of operations to solve for x. Multiplication and Square operation are regular operations. Inverse operations of those are Division and square root operation.

In multi-step equation, the order of solving for x when more than one inverse operation is being involved is

S - subtraction

A - addition

D - division

M - multiplication

E - exponents and roots

P - parenthesis

We divide 2 by both sides of the equation.

So then, we should get x^2 = 1/2, since 2x^2 divided by 2 is x^2 and 1 divided by 2 is 1/2.

x^2 = 1/2

Now, we take the square root of both sides since, we want to get an x on one side of the equation. The only way to undo a square of x in an equation legally is to perform a square root of both sides of the equation.

x = 1 divided by the square root of 2.

if you rationalize the square root of 2 on the bottom, you will get x = square root of 2 over the whole number 2.

Now seeing that the hypothenuse has a numerical value of 1. It makes sense to reference the value of the legs of the right triangle in respect to the degree angle associated with it. The unit circle is completely practical in this case. To find the value of the legs opposite to the angles they carry in respect, we use the value of y for each particular angle.

To distinguish the legs of the right triangle, I use the terms horizontal leg and vertical leg.

Please photo for reference.

- Log in to reply to the answers

- PhilipLv 61 month ago
sin(30°) = (1/2) = opposite/hypotenuse. For hypotenuse = 1, opposite = (1/2). Then

other side length = [1 - (1/2)^2]^(1//2) = (3/4)^(1/2) = (rt3)/2 = (1/2)sqrt3.

- Log in to reply to the answers

- KrishnamurthyLv 71 month ago
Angles: 30°: 60°: 90°

The ratio of sides: 1 : √3 : 2

If the hypotenuse is 1 unit,

the length of the two legs are 1/2 and √3/2.

Now you have an isosceles right triangle, with hypotenuse 1:

a^2 + b^2 = 1.

And since it's isosceles (45 - 45 - 90), a = b, so:

2a^2 = 1

a^2 = 1/2

a = √(1/2) = √2/2

Therefore, both legs equal 1/√2.

- Log in to reply to the answers

- MichaelLv 71 month ago
Looking at the your picture it is NOT a 30-60-90 triangle

It is an isosceles right triangle with a hypotenuse of 1

a² + b² = 1²

Since the legs of an isosceles triangle are equal b² = a²

a² + a² = 1²

2a² = 1

a² = 1/2

Take the square root of both sides

a = √(1/2)

Rationalizing the denominator

the length of the legs is

a = (√2) / 2 <–––––

-------------------------------

-------------------------------

For a 30-60-90 triangle:

If you put two 30-60-90 triangle back to back,

like in the image below, you have an equilateral triangle.

Let the length of the sides be 2x

The short leg, x, of the 30-60-90

which is half the side of the equilateral triangle

is half the length of its hypotenuse, 2x

Using a² + b² = c² the long leg is x√3

For a 30-60-90 triangle the legs are alway the same proportions

short leg = x

long leg = x√3

hypotenuse = 2x

------------------------------

For a hypotenuse of 1,

2x = 1

Short leg = x = 1/2

Long leg = x√3 = (1/2)√3 = (√3) / 2

- Log in to reply to the answers

- What do you think of the answers? You can sign in to give your opinion on the answer.
- Marley KLv 71 month ago
In a 30-60-90 triangle, the legs are 1/2 and √3/2 when the hypotenuse is 1.

Now you have an isosceles right triangle, with hypotenuse 1, but the same principle applies. YES, a^2 + b^2 =1.

since it's isosceles (45-45-90), a=b, so:

2a^2 = 1

a^2 = 1/2

a = √(1/2) = √2/2

therefore, both legs are √2/2.

- Log in to reply to the answers

- 1 month ago
Wait if its isosceles then the legs are congruent so I do 2a^2 equals 1?

I need to get a tutor or something ahhh em confusion

- Log in to reply to the answers