#8) There are THREE right-angle (90°) TRIANGLES.
The "big" triangle has legs "x" and "no designation - <em>so lets call it Z" </em>and hypotenuse "4+5"<em> = 9</em>
The "left" triangle has legs "4" and "y" and hypotenuse "x"
The "right" triangle has legs "5" and "y" and hypotenuse "no designation - <em>which we named Z</em>"
Now, you can create a system of equations - you have three unknown variables and can create an equation for each triangle using the Pythagorean Theorem.
Big: x² + Z² = 9² → x² + Z² = 81
Left: 4² + y² = x² → 16 + y² = x² → 16 = x² - y² → x² - y² = 16
Right: 5² + y² = Z² → 25 + y² = Z²
Next, let's eliminate Z by using substitution....use the Right equation to plug Z² into the Big equation to create a NEW Big equation.
NEW Big: x² + (25 + y² ) = 81 → x² + y² = 81 - 25 → x² + y² = 56
Now, we have two unknown variables (x and y) and two equations (NEW Big and Left). Solve the system using any method. <em>I am going to solve using the elimination method)</em>
NEW Big: x² + y² = 56
+ Left: <u>x² - y² = 16</u>
2x² = 72 <em>added the NEW Big and Left equations</em>
x² = 36 <em>divided both sides by 2</em>
x = 6 <em>took square root of both sides (NOTE: -6 is disregarded because we are solving for a length, which cannot be negative) </em>
The last step is to plug the "x" value (<em>that we just solved for)</em> into one of the equations to solve for "y". <em>I am choosing to use the Left equation.</em>
<em> </em>x² - y² = 16
36 - y² = 16<em> </em>
- y² = 16 - 36
- y² = -20
y² = 20
y = √20
y = 2√5
Answer: x = 6, y = 2√5
Try #9 on your own and then check your answers below:
Big: Z² + x² = 12², Left: 3² + y² = Z², Right: y² + 9² = x²
Answer: x = 6√3, y = 3√3
and plugging the result into 
.
