Chapter 8: Q39. (page 289)

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

Short Answer

Expert verified

Value of,

$\begin{array}{l} x = 4 \\ y = 2 \sqrt{5} \\ z = 3 \sqrt{5} \end{array}$

Step by step solution

Step 1. Given information.

The given figure is,

Step 2. Concept used.

The small right triangle with hypotenuse $x + 2$ is similar to the large right triangle with hypotenuse $2 x + 1$ .

So,

role="math" localid="1648873824994" $\begin{matrix} \frac{x + 2}{2 x + 1} = \frac{x}{x + 2} \\ {(x + 2)}^{2} = x (2 x + 1) [multiplyeachsideby(x+2) (2 x + 1)] \\ x^{2} + 4 + 4 x = 2 x^{2} + x \\ x^{2} - 3 x - 4 = 0 [subtract x^{2} + 4 + 4 x fromeachside] \\ x = \frac{- (- 3) \pm \sqrt{{(- 3)}^{2} - 4 (1) (- 4)}}{2 (1)} \\ x = \frac{3 + 5}{2} [lengthispositive] \\ x = 4 \end{matrix}$

Step 3. Use geometric mean concept.

$y$ is the geometric mean of $x = 4$ and $x + 1 = 5$ .

$\begin{matrix} y = \sqrt{4 (5)} \\ = 2 \sqrt{5} \end{matrix}$

The small right triangle with hypotenuse $z$ is similar to the large right triangle with hypotenuse $2 x + 1 = 9$ .

So,

$\begin{matrix} \frac{z}{9} = \frac{x + 1}{z} \\ z^{2} = 9 (x + 1) [multiplyeachsideby 9 z] \\ z = \sqrt{9 (x + 1)} [takesquarerooteachside] \\ z = 3 \sqrt{5} \end{matrix}$

Therefore, the value of,

$\begin{matrix} x = 4 \\ y = 2 \sqrt{5} \\ z = 3 \sqrt{5} \end{matrix}$

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

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Concept used.

Step 3. Use geometric mean concept.

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Find the values of x,y,and z.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Concept used.

Step 3. Use geometric mean concept.

One App. One Place for Learning.

Most popular questions from this chapter

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