What is the value of x?


A. 6

B. 6√2

C. 12

D. 12√2

enter image source here .

2 Answers
May 26, 2017

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

May 26, 2017

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.