Question

What is the value of x?

A. 6

B. 6√2

C. 12

D. 12√2

.

Answer 1

`12 = x`, or C

Explanation:

We know an angle in the triangle and we know the opposite side. What we want to know is the hypotenuse length. There are two ways to do this, the first one is used for any problem like this, but the second simply comes about since we are dealing with a 45 degree angle.

The first technique uses `sin(theta)` to relate the opposite side length to the hypotenuse:

`sin(theta) = "opposite"/"hypotenuse"`

`theta` is equal to 45 in this case, hypotenuse, or the longest side of the triangle, is `x`, and opposite, or the side opposite the angle, is `6sqrt(2)`. So now we plug everything in and solve for `x`.

`sin(45) = (6sqrt(2))/x` Now we multiply both sides by `x`
`color(red)(x)sin(45) = (color(red)(x)*6sqrt(2))/x`
`xsin(45) = 6sqrt(2)` Divide both sides by `sin(45)`
`(xsin(45))/color(red)(sin(45)) = (6sqrt(2))/color(red)(sin(45))`
`x = (6sqrt(2))/sin(45)=12`

Your answer is C. But let's explore the second method.

45/45/90 right triangles have a special interaction where both side legs (those not the hypotenuse) are identical. This is proven with interactions with `tan(theta)`. Using Pythagorean theorem, we can determine the side length by doubling the square of one side, or:

`(6sqrt(2))^2 + (6sqrt(2))^2 = x^2`
`72*2 = x^2`
`144 = x^2`
`12 = x`

Answer 2

`x=12`

Explanation:

In a triangle with angles `45,45,90`, there will always be two shorter sides with length `a` and the hypotenuse with length `asqrt2`.

The image shows us that the shorter sides have length `6sqrt2`, so the hypotenuse will have length `(6sqrt2)sqrt2=12`.