Section 1
Similarity in Right Triangles
Simplify.$$\sqrt{12}$$
Simplify.$$\sqrt{72}$$
Simplify.$$\sqrt{45}$$
Simplify.$$\sqrt{75}$$
Simplify.$$\sqrt{800}$$
Simplify.$$\sqrt{54}$$
Simplify.$$9 \sqrt{40}$$
Simplify.$$4 \sqrt{28}$$
Simplify.$$\sqrt{30} \cdot \sqrt{6}$$
Simplify.$$\sqrt{5} \cdot \sqrt{35}$$
Simplify.$$\sqrt{\frac{3}{7}}$$
Simplify.$$\sqrt{\frac{9}{5}}$$
Simplify.$$\frac{18}{\sqrt{3}}$$
Simplify.$$\frac{24}{3 \sqrt{2}}$$
Simplify.$$\frac{\sqrt{15}}{3 \sqrt{45}}$$
Find the geometric mean between the two numbers.$$2 \text { and } 18$$
Find the geometric mean between the two numbers.$$3 \text { and } 27$$
Find the geometric mean between the two numbers.$$49 \text { and } 25$$
Find the geometric mean between the two numbers.$$1 \text { and } 1000$$
Find the geometric mean between the two numbers.$$16 \text { and } 24$$
Find the geometric mean between the two numbers.$$22 \text { and } 55$$
Refer to the figure.If $L M=4$ and $M K=8,$ find $J M.$
Refer to the figure.If $L M=6$ and $J M=4,$ find $M K.$
Refer to the figure.If $J M=3$ and $M K=6,$ find $L M$
Refer to the figure.If $J M=4$ and $J K=9,$ find $L K$
Refer to the figure.If $J M=3$ and $M K=9,$ find $L J$
Refer to the figure.If $J M=3$ and $J L=6,$ find $M K$
Refer to the figure.If $J L=9$ and $J M=6,$ find $M K$
Refer to the figure.If $L K=3 \sqrt{6}$ and $M K=6,$ find $J M$
Refer to the figure.If $L K=7$ and $M K=6,$ find $J M$
Find the values of $x, y,$ and $z.$
Prove Theorem $8-1.$
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 $a^{2}+b^{2}=c^{2}$ .
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.
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.
Given: $P Q R S$ is a rectangle;$$\angle A \text { is a right angle. }$$Prove: $B S \cdot R C=P S \cdot Q R=(P S)^{2}$
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.)