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# Chapter 11: Radicals and Geometry Connections

Difficulty Level: Basic Created by: CK-12

## Introduction

Radicals in mathematics are important. By using radicals as inverse operations to exponents, you can solve almost any exponential equation. Radicals such as the square root have been used for thousands of years. Square roots are extremely useful in geometry when finding the hypotenuse of a right triangle or solving for the side length of a square.

In this chapter you will learn the basics of radicals and apply these basics to geometry concepts, such as Pythagorean’s Theorem, the Distance Formula, and the Midpoint Formula. The last several Concepts of this chapter will discuss data analysis, a method used to analyze data by creating charts and graphs.

## Summary

This chapter begins by talking about radicals, including graphing square root functions, simplifying radical expressions, adding and subtracting radicals, multiplying and dividing radicals, and solving radical equations. Next, the Pythagorean Theorem and its converse are discussed, as are the distance formula and the midpoint formula. Finally, the chapter concludes by covering central tendency and dispersion, and it provides instruction on analyzing data with stem-and-leaf plots, histograms, and box-and-whisker plots.

### Radicals and Geometry Connections; Data Anaylsis Review

Explain the shift of each function from the parent function $f(x)=\sqrt{x}$.

1. $f(x)=\sqrt{x}+7$
2. $f(x)=\sqrt{x+3}$
3. $g(x)=-\sqrt{x}$
4. $y=3+\sqrt{x-1}$

Graph the following square root functions. Identify the domain and range of each.

1. $f(x)=\sqrt{x-2}+5$
2. $g(x)=-\sqrt{x+1}$
3. $f(x)=\sqrt{2x}-2$

Simplify the following, if possible. Write your answer in its simplest form.

1. $\sqrt{\frac{3}{7}} \times \sqrt{\frac{14}{27}}$
2. $\sqrt{5} \cdot \sqrt{7}$
3. $\sqrt{11} \times \sqrt[3]{11}$
4. $\frac{\sqrt{18}}{\sqrt{2}}$
5. $8\sqrt[3]{4}+11\sqrt[3]{4}$
6. $5\sqrt{80}-12\sqrt{5}$
7. $\sqrt{10}+\sqrt{2}$
8. $\sqrt{24}-\sqrt{6}$
9. $\sqrt[3]{27}+\sqrt[4]{81}$
10. $4\sqrt{3} \cdot 2\sqrt{6}$
11. $\sqrt[3]{3} \times \sqrt{7}$
12. $6\sqrt{72}$
13. $7\sqrt{\left (\frac{40}{49} \right )}$
14. $\frac{5}{\sqrt{75}}$
15. $\frac{\sqrt{45}}{\sqrt{5}}$
16. $\frac{3}{\sqrt[3]{3}}$
17. $8\sqrt{10}-3\sqrt{40}$
18. $\sqrt{27}+\sqrt{3}$

Solve each equation. If the answer is extraneous, say so.

1. $8=\sqrt[3]{2k}$
2. $x=\sqrt{7x}$
3. $\sqrt{2+2m}=\sqrt{4-m}$
4. $\sqrt[4]{35-2x}=-1$
5. $14=6+\sqrt{10-6x}$
6. $4+\sqrt{\frac{n}{3}}=5$
7. $\sqrt{-9-2x}=\sqrt{-1-x}$
8. $-2=\sqrt[3]{t-6}$
9. $5\sqrt{10}=6\sqrt{w}$
10. $\sqrt{x^2+3x}=2$
11. $\sqrt[4]{t}=5$
12. A leg of a right triangle is 11. The triangle's hypotenuse is 32. What is the length of the other leg?
13. Can 9, 12, 15 be sides of a right triangle?
14. Two legs of a right triangle have lengths of 16 and 24. What is the length of the hypotenuse?
15. Can 20, 21, and 29 be the lengths of the sides of a right triangle?

Find the distance between the two points. Then find the midpoint.

1. (0, 2) and (–5, 4)
2. (7, –3) and (4, –3)
3. (4, 6) and (–3, 0)
4. (8, –3) and (–7, –6)
5. (–8, –7) and (6, 5)
6. (–6, 6) and (0, 8)
7. (2, 6) is six units away from a second point. Find the two possibilities for this ordered pair.
8. (9, 0) is five units away from a second point. Find the two possibilities for this ordered pair.
9. The midpoint of a segment is (7.5, 1.5). Point $A$ is (–5, –6). Find the other endpoint of the segment.
10. Maggie started at the center of town and drove nine miles west and five miles north. From this location, she drove 16 miles east and 12 miles south. What is the distance from this position from the center of town? What is the midpoint?
11. The surface area of a cube is given by the formula $SA=6s^2$. The surface area is 337.50 square inches. What is the side length of the cube?
12. The diagonal of a sail is 24 feet long. The vertical length is 16 feet. If the area is found by $\frac{1}{2} (length)(height)$, determine the area of the sail.
13. A student earned the following test scores: 63, 65, 80, 84, 73. What would the next test score have to be in order to have an average of 70?
14. Find the mean, median, mode, and range of the data set. 11, 12, 11, 11, 11, 13, 13, 12, 12, 11, 12, 13, 13, 12, 13, 11, 12, 12, 13
15. A study shows the average teacher earns $45,000 annually. Most teachers do not earn close to this amount. 1. Which central tendency was most likely used to describe this situation? 2. Which measure of central tendency should be used to describe this situation? 1. Mrs. Kramer’s Algebra I class took a test on factoring. She recorded the scores as follows: 55, 57, 62, 64, 66, 68, 68, 68, 69, 72, 75, 77, 78, 79, 79, 82, 83, 85, 88, 90, 90, 90, 90, 92, 94, 95, 97, 99 1. Construct a histogram using intervals of ten, starting with 50–59. 2. What is the mode? What can you conclude from this graph? 1. Ten waitresses counted their tip money as follows:$32, $58,$17, $27,$69, $73,$42, $38,$24, and \$52. Display this information in a stem-and-leaf plot.
2. Eleven people were asked how many miles they live from their place of work. Their responses were: 5.2, 18.7, 8.7, 9.1, 2.3, 2.3, 5.4, 22.8, 15.2, 7.8, 9.9. Display this data as a box-and-whisker plot.
3. What is one disadvantage of a box-and-whisker plot?
4. Fifteen students were randomly selected and asked, “How many times have you checked Facebook today?” Their responses were: 4, 23, 62, 15, 18, 11, 13, 2, 8, 7, 12, 9, 14, 12, 20. Display this information as a box-and-whisker plot and interpret its results.
5. What effect does an outlier have on the look of a box-and-whisker plot?
6. Multiple Choice. The median always represents which of the following? A. The upper quartile B. The lower quartile C. The mean of the data D. The 50% percentile

### Radicals and Geometry Connections; Data Anaylsis Test

1. Describe each type of visual display presented in this chapter. State one advantage and one disadvantage for each type of visual display.
2. Graph $f(x)=7+\sqrt{x-4}$. State its domain and range. What is the ordered pair of the origin?
3. True or false? The upper quartile is the mean of the upper half of the data.
4. What is the domain restriction of $y=\sqrt[4]{x}$?
5. Solve $-6=2\sqrt[3]{c+5}$.
6. Simplify $\frac{4}{\sqrt{48}}$.
7. Simplify and reduce: $\sqrt[3]{3} \times \sqrt[3]{81}$.
8. A square baking dish is 8 inches by 8 inches. What is the length of the diagonal? What is the area of a piece cut from corner to opposite corner?
9. The following data consists of the weights, in pounds, of 24 high school students: 195, 206, 100, 98, 150, 210, 195, 106, 195, 108, 180, 212, 104, 195, 100, 216, 99, 206, 116, 142, 100, 135, 98, 160.
1. Display this information in a box plot, a stem-and-leaf plot, and a histogram with a bin width of 10.
2. Which graph seems to be the best method to display this data?
3. Are there any outliers?
1. Find the distance between (5, –9) and (–6, –2).
2. The coordinates of Portland, Oregon are (43.665, 70.269). The coordinates of Miami, Florida are (25.79, 80.224).
1. Find the distance between these two cities.
2. What are the coordinates of the town that represents the halfway mark?
1. The Beaufort Wind Scale is used by coastal observers to estimate the wind speed. It is given by the formula $s^2=3.5B^3$, where $s=$ the wind speed (in knots) and $B=$ the Beaufort value.
1. Find the Beaufort value for a 26-knot wind.
2. What is the wind speed of a severe storm with a gale wind of 50 knots?
1. Find the two possibilities for a coordinate ten units away from (2, 2).
2. Use the following data obtained from the American Veterinary Medical Association. It states the number of households per 1,000 with particular exotic animals.
1. Find the mean, median, mode, range, and standard deviation.
2. Are there any outliers? What effect does this have on the mean and range?
Households
(in 1,000)
Fish 9,036
Ferrets 505
Rabbits 1,870
Hamsters 826
Guinea Pigs 628
Gerbils 187
Other Rodents 452
Turtles 1,106
Snakes 390
Lizards 719

#### Texas Instruments Resources

In the CK-12 Texas Instruments Algebra I FlexBook, there are graphing calculator activities designed to supplement the objectives for some of the lessons in this chapter. See http://www.ck12.org/flexr/chapter/9621.

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Feb 24, 2012