Exercise 50 Page 544

We are given a figure and asked to evaluate the values of $x,y$ and $x.$

To do this, we will use the Right Triangle Geometric Mean Theorems. Let's start with evaluating the value of $x.$

$x$

Let's recall that according to the Geometric Mean Leg Theorem the length of the leg of a right triangle is the geometric mean between the length of the hypotenuse and the segment of the hypotenuse adjacent to that leg.

We can apply this theorem to our exercise. Notice that the length of the hypotenuse of the drawn triangle will be $x+8.8.$ Let's solve this equation for $x.$ Next, we will use the Quadratic Formula to solve the given quadratic equation. We first need to identify the values of $a,$ $b,$ and $c.$ We see that $a=1,$ $b=8.8,$ and $c=\text{-}72.25.$ Let's substitute these values into the Quadratic Formula. Since $x$ cannot be a negative number, we will consider only the positive case. The value of $x$ is approximately $5.17.$ Let's add this information to our picture.

$y$

Now, we will evaluate the value of $y.$ According to the Geometric Mean Altitude Theorem, the altitude drawn to the hypotenuse of a right triangle separates the hypotenuse into two segments. The length of this altitude is the geometric mean between the lengths of these two segments.

We can create and solve an equation according to the theorem we recalled above. The value of $y$ is approximately $6.75.$ We can add this information to our picture.

$z$

To find the value of $z,$ we will use the same theorem as we used to find $x.$ According to this theorem, the length of the leg of a right triangle is the geometric mean between the length of the hypotenuse and the segment of the hypotenuse adjacent to that leg.

Let's create and solve an equation using the above theorem. The value of $z$ is approximately $11.09.$