Find x.
math question no guessing

In the bigger triangle, we know two of the angles (90 degrees and 45 degrees). Therefore, we can work out the remaining angle by doing 180 - (90 + 45) = 45 degrees. (Because all the angles in a triangle add up to 180 degrees).

Now that we know all three angles, we can see that two of the angles are the same (45 and 45), which means the bigger triangle is an isosceles triangle. This means two of the sides are the same. Therefore, when we use Pythagoras’ theorem to work out the side lengths, we can treat a^2 and b^2 as the same value (so it’d be 2a^2, or even 2b^2). Now we can work it out as we know the hypotenuse is 6.

2a^2 = 6^2
2a^2 = 36
a^2 = 18
a = √18 = 3√2

Now we know that the remaining side lengths of the big triangle are both 3√2.

That also means that the longest side of the smaller triangle is also 3√2 (as shown on the diagram), which will be useful later on.

Now we should work out all the angles of the smaller triangle, and as you know, all the angles add up to 180, so you do 180 - (90 + 60) = 30 degrees. Now we can find out x by using the sine rule.

Just do (3√2)/sin(90) * sin(30) to give you the answer. This should equal (3√2)/2 and that should be your final answer, which is A on your multiple choice list.

Hope that helps! :)