Question

Solve for x and y. Show your work.

Answer 1

Answer:

• x = 3
• y = 3√2

Step-by-step explanation:

We can find the value of x and y of the triangles using the ratio of 45°-45°-90° triangle and 30°-60°-90° triangle.

Characteristics of a 45°-45°-90° Triangle:

• It is an isosceles right triangle, which means both perpendicular legs are same length.
• Using the Pythagorean Formula:

       (let the length of the leg = a)

       $hypotenuse=\sqrt{a^2+a^2}$

                          $=\sqrt{2a^2}$

                          $=a\sqrt{2}$

      $\boxed{\bf leg:hypotenuse=1:\sqrt{2} }$

Characteristics of a 30°-60°-90° Triangle:

• It is half of an equilateral triangle, which means the hypotenuse is twice the length of the short perpendicular legs.
• Using the Pythagorean Formula:

       (let the length of the short leg = a, then the hypotenuse = 2a)

       $hypotenuse=\sqrt{short\ leg^2+long\ leg^2}$

                     $2a=\sqrt{a^2+long\ leg^2}$

                 $(2a)^2=a^2+long\ leg^2$  

            $long\ leg =\sqrt{4a^2-a^2}$

            $long\ leg =\sqrt{3a^2}$

            $long\ leg =a\sqrt{3}$

           $\boxed{\bf short\ leg:long\ leg:hypotenuse=1:\sqrt{3} :2 }$

For the 30°-60°-90° Triangle, given hypotenuse = 6

$short\ leg:hypotenuse=1:2$

                           $x:6=1:2$

                              $2x=6$

                                $\bf x=3$

For the 45°-45°-90° Triangle, given hypotenuse = 6

$leg:hypotenuse=1:\sqrt{2}$

                  $y : 6 = 1:\sqrt{2}$

                  $\sqrt{2} y=6$

                       $\displaystyle y=\frac{6}{\sqrt{2} }$

                       $\bf y=3\sqrt{2}$