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Geometry

Ray C. Jurgensen, Richard G. Brown, John W. Jurgensen

$Math Studyset Vaia Explanations$ Math

Student Edition · 227 pages

ISBN: 9780395977279

Chapter 8: Q38. (page 289)

Find the values of $x, y,$ and $z$ .

Short Answer

Expert verified

Value of,

$\begin{matrix} x = 9 \\ y = 15 \\ x = 20 \end{matrix}$

Step by step solution

01

Step 1. Given information.

The given figure is,

02

Step 2. Use geometric mean concept.

$12$ is the geometric mean of $x$ and $x + 7$ .

$\begin{matrix} 12 = \sqrt{x (x + 7)} \\ 12^{2} = x^{2} + 7 x [squareeachside] \\ x^{2} + 7 x - 144 = 0 [subtract144fromeachside] \\ x = \frac{- 7 \pm \sqrt{7^{2} - 4 (1) (- 144)}}{2 (1)} \\ x = \frac{- 7 \pm \sqrt{625}}{2} \\ x = \frac{- 7 \pm 25}{2} \\ x = \frac{- 7 + 25}{2} [sincelengthispositive] \\ \Rightarrow x = 9 \end{matrix}$

03

Step 3. Concept used.

The small right triangle with hypotenuse $y$ is similar to the large right triangle with hypotenuse $2 x + 7$ .

So,

$\begin{matrix} \frac{y}{2 x + 7} = \frac{x}{y} \\ y^{2} = x (2 x + 7) [multiplyeachsidebyy (2 x + 7)] \\ y^{2} = 9 (2 (9) + 7) \\ y = \sqrt{9 \cdot 25} [takesquarerooteachside] \\ y = 15 \end{matrix}$

The small right triangle with hypotenuse $z$ is similar to the large right triangle with hypotenuse $2 x + 7 = 25$ .

So,

$\begin{matrix} \frac{z}{25} = \frac{x + 7}{z} \\ z^{2} = 25 (9 + 7) [multiplyeachsideby25z] \\ z^{2} = 25 (16) \\ z = 20 [takesquarerooteachside] \end{matrix}$

Therefore, the value of,

$\begin{matrix} x = 9 \\ y = 15 \\ z = 20 \end{matrix}$

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Most popular questions from this chapter

The arithmetic mean between two numbers $r$ and $s$ is defined to be $\frac{r + s}{2}$ .

$(a) .$ $\bar{C M}$ is the median and $\bar{C H}$ is the altitude to the hypotenuse of right $Δ A B C$ . Show that $C M$ is the arithmetic mean between $A H$ and $B H$ , and that $C H$ is the geometric mean between $A H$ and $B H$ . Then use the diagram to show that the arithmetic mean is greater than the geometric mean.

$(b) .$ Show algebraically that the arithmetic mean between two different numbers $r$ and $s$ is greater than the geometric mean.

$(a) .$

Simplify

$\frac{\sqrt{5}}{\sqrt{2}}$

Use the diagram to complete each statement..

$\begin{array}{l} I f m ∠ R = 30, t h e n m ∠ R S U = ?, \\ m ∠ T S U = ?, a n d m ∠ T = ? \end{array}$

State an equation you could use to find the value of $x$ . Then find the value of $x$ in simplest radical form.

Refer to the figure at the right.

If $L M = 4$ and $M K = 8$ , find $J M$ .

See all solutions

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