Chapter 8: Q37. (page 289)

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

Short Answer

Expert verified

Value of,

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

Step by step solution

Step 1. Given information.

The given figure is,

Step 2. Concept used.

The small right triangle with hypotenuse $2$ is similar to the large right triangle with hypotenuse $\sqrt{2} + x$ .

So,

$\begin{matrix} \frac{2}{\sqrt{2} + x} = \frac{\sqrt{2}}{2} \\ 2^{2} = \sqrt{2} (\sqrt{2} + x) [multiplyeachsideby2 (\sqrt{2} + x)] \\ 4 = 2 + \sqrt{2} x \\ \sqrt{2} x = 2 [subtract2fromeachside] \\ x = \frac{2}{\sqrt{2}} [divideeachsideby \sqrt{2}] \\ x = \sqrt{2} \end{matrix}$

Step 3. Use geometric mean concept.

$z$ is the geometric mean of $\sqrt{2}$ and $x$ .

$\begin{matrix} z = \sqrt{\sqrt{2} \cdot x} \\ z = \sqrt{\sqrt{2} \cdot \sqrt{2}} \\ z = \sqrt{2} \end{matrix}$

The small right triangle with hypotenuse $y$ is similar to the large right triangle with hypotenuse $\sqrt{2} + x$ .

So,

$\begin{matrix} \frac{y}{\sqrt{2} + x} = \frac{x}{y} \\ y^{2} = x (\sqrt{2} + x) ................... [multiplyeachsidebyy (\sqrt{2} + x)] \\ y^{2} = \sqrt{2} (\sqrt{2} + \sqrt{2}) \\ y^{2} = \sqrt{2} (2 \sqrt{2}) \\ y^{2} = 4 \\ y = 2................................ [takesquarerooteachside] \end{matrix}$

Therefore, the value of,

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

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

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

Recommended explanations on Math Textbooks

Logic and Functions

Calculus

Mechanics Maths

Probability and Statistics

Discrete Mathematics

Pure Maths

Study anywhere. Anytime. Across all devices.

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

Recommended explanations on Math Textbooks

Logic and Functions

Calculus

Mechanics Maths

Probability and Statistics

Discrete Mathematics

Pure Maths

Study anywhere. Anytime. Across all devices.

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