# Geometry

## Educators KB

### Problem 1

Simplify.
$$\sqrt{12}$$ Amrita B.

### Problem 2

Simplify.
$$\sqrt{72}$$

KB
Karan B.

### Problem 3

Simplify.
$$\sqrt{45}$$ Amrita B.

### Problem 4

Simplify.
$$\sqrt{75}$$

KB
Karan B.

### Problem 5

Simplify.
$$\sqrt{800}$$ Amrita B.

### Problem 6

Simplify.
$$\sqrt{54}$$

KB
Karan B.

### Problem 7

Simplify.
$$9 \sqrt{40}$$ Amrita B.

### Problem 8

Simplify.
$$4 \sqrt{28}$$

KB
Karan B.

### Problem 9

Simplify.
$$\sqrt{30} \cdot \sqrt{6}$$ Amrita B.

### Problem 10

Simplify.
$$\sqrt{5} \cdot \sqrt{35}$$

KB
Karan B.

### Problem 11

Simplify.
$$\sqrt{\frac{3}{7}}$$ Amrita B.

### Problem 12

Simplify.
$$\sqrt{\frac{9}{5}}$$

KB
Karan B.

### Problem 13

Simplify.
$$\frac{18}{\sqrt{3}}$$ Amrita B.

### Problem 14

Simplify.
$$\frac{24}{3 \sqrt{2}}$$

KB
Karan B.

### Problem 15

Simplify.
$$\frac{\sqrt{15}}{3 \sqrt{45}}$$ Amrita B.

### Problem 16

Find the geometric mean between the two numbers.
$$2 \text { and } 18$$

KB
Karan B.

### Problem 17

Find the geometric mean between the two numbers.
$$3 \text { and } 27$$ Amrita B.

### Problem 18

Find the geometric mean between the two numbers.
$$49 \text { and } 25$$

KB
Karan B.

### Problem 19

Find the geometric mean between the two numbers.
$$1 \text { and } 1000$$ Amrita B.

### Problem 20

Find the geometric mean between the two numbers.
$$16 \text { and } 24$$

KB
Karan B.

### Problem 21

Find the geometric mean between the two numbers.
$$22 \text { and } 55$$ Amrita B.

### Problem 22

Refer to the figure.
If $L M=4$ and $M K=8,$ find $J M.$

KB
Karan B.

### Problem 23

Refer to the figure.
If $L M=6$ and $J M=4,$ find $M K.$ Amrita B.

### Problem 24

Refer to the figure.
If $J M=3$ and $M K=6,$ find $L M$

KB
Karan B.

### Problem 25

Refer to the figure.
If $J M=4$ and $J K=9,$ find $L K$ Amrita B.

### Problem 26

Refer to the figure.
If $J M=3$ and $M K=9,$ find $L J$

KB
Karan B.

### Problem 27

Refer to the figure.
If $J M=3$ and $J L=6,$ find $M K$ Amrita B.

### Problem 28

Refer to the figure.
If $J L=9$ and $J M=6,$ find $M K$

KB
Karan B.

### Problem 29

Refer to the figure.
If $L K=3 \sqrt{6}$ and $M K=6,$ find $J M$ Amrita B.

### Problem 30

Refer to the figure.
If $L K=7$ and $M K=6,$ find $J M$

KB
Karan B.

### Problem 31

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

### Problem 32

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

KB
Karan B.

### Problem 33

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

### Problem 34

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

KB
Karan B.

### Problem 35

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

### Problem 36

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

KB
Karan B.

### Problem 37

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

### Problem 38

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

KB
Karan B.

### Problem 39

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

### Problem 40

Prove Theorem $8-1.$

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### Problem 41

Refer to the figure at the right, and use Corollary 2 to complete:
$$a^{2}=\underline{?} \text { and } b^{2}=\underline{?}$$
b. Add the equations in part (a), factor the sum on the right, and show that $a^{2}+b^{2}=c^{2}$ . Amrita B.

### Problem 42

Prove: In a right triangle, the product of the hypotenuse and the length of the altitude drawn to the hypotenuse equals the product of the two legs.

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### Problem 43

Given: $P Q R S$ is a rectangle;
$P S$ is the geometric mean between $S T$ and $T R .$
Prove: $\angle P T Q$ is a right angle. Amrita B.

### Problem 44

Given: $P Q R S$ is a rectangle;
$$\angle A \text { is a right angle. }$$
Prove: $B S \cdot R C=P S \cdot Q R=(P S)^{2}$

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### Problem 45

The arithmetic mean between two numbers $r$ and $s$ is defined to be $\frac{r+s}{2}.$
a. $\overline{C M}$ is the median and $\overline{C H}$ is the altitude to the hypotenuse of right $\triangle A B C$ . 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. (Hint: The geometric mean is $\sqrt{r s} .$ Work backward from $\frac{r+s}{2}>\sqrt{r s}$ to $(r-s)^{2}>0$ and then reverse the steps.)

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