Question

Which three pairs of measurements are possible side length for the triangle?

Answer 1

A, B, E, F

Explanation:

In a 30-60-90 triangle, the hypotenuse is twice the length of the short leg.

That makes choice E possible.

In a 30-60-90 triangle, the long leg is sqrt(3) times the length of the short leg.

That makes choices A, B, and F possible.

Answer 2

First option.

Option 5.

Option 6.

Explanation:

The formula for a 30-60-90 triangle is this:

1) Side opposite to 30 will be value
`a`.

2) Side opposite to 60 will be value
`a√(3)`.

3) Hypotenuse will be
`2a`.

So let's look and see:

First option:
`AB=4` and
`BC=4√(3)`

AB is opposite of the angle with 30 degree measurement.

BC is opposite of the angle with 60 degree measurement.

So
`a=4` here.

So the side opposite of 60 using the formula should be
`4 √(3)` which it is here.

So first option looks good.

Second option:
`BC=2√(3)` and
`AC=2`.

We aren't given the side opposite to 30.

AC is the hypotenuse so 2a=2 which means the side opposite to 30 is a=2/2=1.

This means using the formula that the side opposite to 60 will be
`1√(3)=√(3)` but we don't have that.

So not option 2.

Third option:
`AB=3` and
`AC=3√(3)`

AB is the side opposite of 30, so we have
`a=3`

AC is the hypotenuse so that side should be
`2a=6` and it isn't.

Option 3 is not working.

Fourth option:
`BC=10` and
`AC=4√(3)`

So we have that
`2a=4√(3)` which means
`a=2√(3)` and so
`a√(3)=2√(3)√(3)=2(3)=6` but that is a contradiction because we have this value should be 10.

Not option 4.

Option 5:
`AB=7` and
`AC=14`

So we have
`a=7` and
`2a=14` so this looks good.

Option 6:
`AB=11` and
`BC=11√(3)`

`a=11` so
`a√(3)=11√(3)` which is what we have.

Option 6 works.