A 30°-60°-90° triangle has a shorter leg of length 4. What are the… Step by Step

Step by step

The question

Question: find the other two sides of a 30-60-90 triangle with a short leg of 4.

There are four choices.

First we solve it, the normal way.

Then we pick.

1 Half of an equilateral

The angles are 30°, 60° and 90°.

Flip a copy over the long leg.

It goes under it.

The two 30° angles together make 60°.

So the big triangle has three 60° angles.

Equal angles, so equal sides: an equilateral triangle.

The short leg is across from the 30° angle.

It is half of one side.

We call it x.

The other half of that side is x too.

The hypotenuse is one whole side.

So it is 2 times the short leg.

2 Why times √3

How long is the long leg?

Pythagoras: x² + long² = (2x)².

Take away x² on both sides.

long² = (2x)² − x²

(2x)² is 2x times 2x: 2 times 2 is 4, and x times x is x².

So 4x².

long² = 4x² − x²

4x² take away x² is 3x².

long² = 3x²

The long leg times itself is 3x².

Which number times itself is 3x²?

x√3.

x times x is x².

And √3 times √3 is 3: that is what √3 means.

long = x√3

So the long leg is the short leg times √3.

3 Find the short leg

The short leg is given: x = 4.

x = 4

4 Find the hypotenuse

The hypotenuse is 2x: 2 times 4.

hypotenuse = 2 × 4

So the hypotenuse is 8.

hypotenuse = 8

5 Find the long leg

The long leg is x times √3: 4 times √3.

long leg = 4 × √3

So the long leg is 4√3.

long leg = 4√3

Careful!

√3 is for the long leg of this triangle.

√2 is for the 45-45-90 triangle.

Careful: not 4√2 ✗

So the hypotenuse is 8 and the long leg is 4√3.

hypotenuse 8 and long leg 4√3

6 Check

Let's check with Pythagoras: short leg squared plus long leg squared.

(4√3)² is 4 × 4 times √3 × √3: 16 times 3.

4² + (4√3)² = 16 + 16 × 3 = 16 + 48 = 64

The hypotenuse squared: 8² is 64.

8² = 64

Both are 64.

So the sides are right!

7 Pick the choice

Our answer is hypotenuse 8, long leg 4√3.

That is choice C.

C) hypotenuse 8, long leg 4√3

8 Why not the others

A is hypotenuse 4√2, long leg 4.

Wrong: that is the 45-45-90 rule.

A) hypotenuse 4√2, long leg 4 ✗ that is the 45-45-90 rule

B is hypotenuse 8, long leg 4√2.

Wrong: the long leg is the short leg × √3, not √2.

B) hypotenuse 8, long leg 4√2 ✗ the long leg is the short leg × √3, not √2

D is hypotenuse 8, long leg 8√3.

Wrong: used the hypotenuse × √3: the long leg is the short leg × √3.

D) hypotenuse 8, long leg 8√3 ✗ used the hypotenuse × √3: the long leg is the short leg × √3

Let's look back.

First: half of an equilateral.

Then: why times √3.

Then: find the short leg.

Then: find the hypotenuse.

Then: find the long leg.

Then: check.

Then: pick the choice.

Last: why not the others.

Answer