Chapter 8.1, Problem 17CE

a.

To determine

To fill: the blanks with suitable parameter based on the given triangle.

Answer to Problem 17CE

z is the geometric mean between 2 and 7

Thus $z=\sqrt{14}$

Explanation of Solution

Given Information: A triangle with a right angle and a perpendicular directed to the hypotenuse. The measure of sides is marked.

Formula used:

For a right triangle with a perpendicular being drawn to the hypotenuse (as shown below), the relation between the hypotenuse and the normal to it is expressed as,

  

  $\begin{array}{l}{h}^2=xy\\ h=\sqrt{xy}\end{array}$

This implies the length of the normal is the geometric mean of the two parts of the hypotenuse divided by it.

Calculation :

Consider the triangle shown below.

  

For the given triangle, the altitude is represented by zand the parts of the hypotenuse are 2 and 7. Thus, comparing with the general figure and formula, the length z can be calculated as,

  $\begin{array}{c}z=\sqrt{r\times s}\\ =\sqrt{2\times 7}\end{array}$

This implies z is the geometric mean of 2 and 7.

The value of z is given by,

  $\begin{array}{c}z=\sqrt{2\times 7}\\ =\sqrt{14}\end{array}$

b.

To determine

To fill: the blanks for the given statement

Answer to Problem 17CE

x is the geometric mean between 2 and 9

Thus $x=3\sqrt{2}$

Explanation of Solution

Given Information: A triangle with a right angle and a perpendicular directed to the hypotenuse. The measure of sides is marked.

Formula used:

For a right triangle with a perpendicular being drawn to the hypotenuse (as shown below),the both small triangles are congruent to each other and to the bigger triangle. This gives the relation between the hypotenuse and one of the legs of the smaller triangle as,

  

  $\begin{array}{l}{b}^2=cx\\ b=\sqrt{cx}\end{array}$

Calculation:

Consider the triangle shown below.

  

For the given triangle, the one of the legs of the bigger triangle is represented by x and the parts of the hypotenuse considered is 2 and the total length of hypotenuse being 9. Thus, comparing with the general figure and formula, the value of x can be expressed as,

  $x=\sqrt{2\times 9}$

Thus, x is the geometric mean of 2 and 9.

The value of x is given by,

  $\begin{array}{c}x=\sqrt{2\times 9}\\ =\sqrt{18}\\ =3\sqrt{2}\end{array}$

c.

To determine

To fill: the blanks for the given statement

Answer to Problem 17CE

y is the geometric mean between 7 and 9 Thus $y=3\sqrt{7}$

Explanation of Solution

Given Information: A triangle with a right angle and a perpendicular directed to the hypotenuse. The measure of sides is marked.

Formula used:

For a right triangle with a perpendicular being drawn to the hypotenuse (as shown below), both small triangles are congruent to each other and to the bigger triangle. This gives the relation between the hypotenuse of the bigger triangle and one of the legs of smaller triangle as,

  

  $\begin{array}{l}{a}^2=cy\\ a=\sqrt{cy}\end{array}$

Calculation:

Consider the triangle shown below.

  

For the given triangle, the one of the legs of the bigger triangle is represented by y and the parts of the hypotenuse considered is 7 and the total length of hypotenuse being 9. Thus, comparing with the general figure and formula, the value of y can be calculated as,

  $y=\sqrt{7\times 9}$

Thus, y is the geometric mean of 7 and 9.

The value of y is given by,

  $\begin{array}{c}y=\sqrt{7\times 9}\\ =\sqrt{63}\\ =3\sqrt{7}\end{array}$