Question

The hypotenuse of a 30°-60°-90º triangle is 8 units and one leg is 4 units. What is the length of the other leg?

Answer 1

Answer:

The length of the other leg is $4\sqrt{3}$

Step-by-step explanation:

There are 2 ways to solve this.

1) Pythagorean theorem (the harder way)

A 30-60-90 triangle is a right triangle, and therefore you can use the pythagorean theorem to calculate the measure of 1 side given the 2 others.

$a^2+b^2=c^2\\a^2+4^2=8^2\\a^2+16=64\\a^2=48\\a=\sqrt{48}\\a=6.9282...$

That's one solution, but its not a very nice number or the exact answer you probably want. Instead, you can simplify the root.

First, you need to factor 48:

$2*24=48\\2*12=24\\2*6=12\\2*3=6$

The prime factors of 48 are:

$2*2*2*2*3=48$

Then, you need to find perfect squares:

$(2*2)*(2*2)*3=48\\2^2*2^2*3=48$

And finally, you can remove those like this:

$\sqrt{48} =\sqrt{2^2*2^2*3} \\\sqrt{48} =2\sqrt{2^2*3} \\\sqrt{48} =2*2\sqrt{3} \\\sqrt{48} =4\sqrt{3}$

The length of the other leg is $4\sqrt{3}$

2) 30-60-90 triangle relationships (the easier way)

There are 4 simple rules here:

• Short leg × 2 = hypotenuse
• Short leg × √3 = long leg
• Long leg ÷ √3 = short leg
• Hypotenuse ÷ 2 = short leg

They're actually pretty easy to remember. You can multiply the short leg by 2 or √3, and 2 is greater than √3 (check a calculator). The hypotenuse of course is also longer than the long leg, so you have to multiply by 2 to get it. Then for the long leg, the only option is to multiply by √3. The same works in reverse for both of those.

Here, we have the short leg. You can check that with the pythagorean theorem again to be sure:

$a^2+4^2=8^2\\a^2+16=64$

Because 16 is less than half of 64, a has to be the longer leg. Using the rules above, multiply the short leg by √3 to get the answer:

$4 * \sqrt{3} \\4\sqrt{3}$

Same answer as the previous method.