Question

Find the value of x and the value of y

Answer 1

Answer: Hello there!

You can see that this is a triangle rectangle, where one of the catheti is equal to 5, and we know all the angles (you can see that both lower angles are equal to 45°, then is safe to assume that both catheti are equal in length, but let's do the math):

we need to find the length of the other cathetus and the length of the hypotenuse.

A very useful thing to remember when you are working with a triangle rectangle is:

So-Ca -Toa

this means:

sin(a) = (opposite cathetus)/(hipotenuse)

cos(a) = (adjacent cathetus)/(hipotenuse)

tan(a) = (opposite cathetus)/(adjacent cathetus)

now, we can find y using the third relation:

tan(45°) = 5/x

y= 5/tan(45°) = 5

because tan(45°) = 1

and this has a lot of sence, because you can see that both cathetus are symetric.

Now we can find the value of x with the next Pythagorean theorem:

the square hypotenuse is equal to the sum of the squares of the cathetus:

x^2 = 5^2 + 5^2 = 2*(5^2)

x= (√2)*5

then the right answer is option B:

x = (√2)*5 and y = 5

Answer 2

It's an isosceles triangle 45° - 45° - 90°, therefore y = 5.

In that triangle, sides are in proportion 1 : 1 : √2.

Therefore we have x = 5√2.

Answer: B. x = 5√2, y = 5

Other method.

You can use the Pythagorean theorem:

x² = 5² + 5²

x² = 25 + 25

x² = 2 · 25

x = √(25 · 2)

x = √25 · √2

x = 5√2

Answer: B. x = 5√2, y = 5