Chapter 9 Review and Practice Test

Learning objectives

Learning objectives

• Review the key ideas of this chapter.

• Apply the chapter's methods to review exercises.

• Check your understanding with a practice test.

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Worked example

angle

An angle is formed by two rays that share a common endpoint. Each ray is called a side of the angle.

area

The area is a measure of the surface covered by a figure.

complementary angles

cone

A cone is a solid figure with one circular base and a vertex.

cube

A cube is a rectangular solid whose length, width, and height are equal.

cylinder

A cylinder is a solid figure with two parallel circles of the same size at the top and bottom.

equilateral triangle

A triangle with all three sides of equal length is called an equilateral triangle.

hypotenuse

The side of the triangle opposite the 90° angle is called the hypotenuse.

irregular figure

An irregular figure is a figure that is not a standard geometric shape. Its area cannot be calculated using any of the standard area formulas.

isosceles triangle

A triangle with two sides of equal length is called an isosceles triangle.

legs of a right triangle

The sides of a right triangle adjacent to the right angle are called the legs.

perimeter

The perimeter is a measure of the distance around a figure.

rectangle

A rectangle is a geometric figure that has four sides and four right angles.

right triangle

similar figures

In geometry, if two figures have exactly the same shape but different sizes, we say they are similar figures.

supplementary angles

trapezoid

A trapezoid is four-sided figure, a quadrilateral, with two sides that are parallel and two sides that are not.

triangle

A triangle is a geometric figure with three sides and three angles.

vertex of an angle

When two rays meet to form an angle, the common endpoint is called the vertex of the angle.

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Key Concepts

• Problem Solving StrategyStep 1. Read the word problem. Make sure you understand all the words and ideas. You may need to read the problem two or more times. If there are words you don't understand, look them up in a dictionary or on the internet. Step 2. Identify what you are looking for. Step 3. Name what you are looking for. Choose a variable to represent that quantity. Step 4. Translate into an equation. It may be helpful to first restate the problem in one sentence before translating. Step 5. Solve the equation using good algebra techniques. Step 6. Check the answer in the problem. Make sure it makes sense. Step 7. Answer the question with a complete sentence.

• Finding the Total Value for Coins of the Same Type For coins of the same type, the total value can be found as follows: $\begin{array}{l} \mathrm{number}\cdot\mathrm{value}=\mathrm{total value} \\ \end{array}$where number is the number of coins, value is the value of each coin, and total value is the total value of all the coins.

• Solve a Coin Word ProblemStep 1. Read the problem. Make sure you understand all the words and ideas, and create a table to organize the information. Step 2. Identify what you are looking for. Step 3. Name what you are looking for. Choose a variable to represent that quantity.Use variable expressions to represent the number of each type of coin and write them in the table. Multiply the number times the value to get the total value of each type of coin. Step 4. Translate into an equation. Write the equation by adding the total values of all the types of coins. Step 5. Solve the equation using good algebra techniques. Step 6. Check the answer in the problem and make sure it makes sense. Step 7. Answer the question with a complete sentence.

• Type $\mathrm{Number}$$\mathrm{Value (\textdollar )}$$\mathrm{Total Value (\$)}$

• Supplementary and Complementary AnglesIf the sum of the measures of two angles is 180°, then the angles are supplementary. If $∠A$ and $∠B$ are supplementary, then $m∠A+m∠B=180$. If the sum of the measures of two angles is 90°, then the angles are complementary. If $∠A$ and $∠B$ are complementary, then $m∠A+m∠B=90$.

• Solve Geometry ApplicationsStep 1. Read the problem and make sure you understand all the words and ideas. Draw a figure and label it with the given information. Step 2. Identify what you are looking for. Step 3. Name what you are looking for and choose a variable to represent it. Step 4. Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information. Step 5. Solve the equation using good algebra techniques. Step 6. Check the answer in the problem and make sure it makes sense. Step 7. Answer the question with a complete sentence.

• Sum of the Measures of the Angles of a TriangleFor any $\Delta{}ABC,$ the sum of the measures is 180° $m∠A+m∠B+m∠C=180$

• Right Triangle A right triangle is a triangle that has one 90° angle, which is often marked with a $\measuredangle$symbol.

• Properties of Similar TrianglesIf two triangles are similar, then their corresponding angle measures are equal and their corresponding side lengths have the same ratio.

• Properties of Rectangles Rectangles have four sides and four right (90°) angles. The lengths of opposite sides are equal. The perimeter, $P$, of a rectangle is the sum of twice the length and twice the width. $P=2L+2W$ The area, $A$, of a rectangle is the length times the width. $A=L\cdot{}W$

• Triangle Properties For any triangle $\Delta{}ABC$, the sum of the measures of the angles is 180°. $m\mathrm{∠}A+m\mathrm{∠}B+m\mathrm{∠}C=180^{\circ}$ The perimeter of a triangle is the sum of the lengths of the sides. $P=a+b+c$ The area of a triangle is one-half the base, b, times the height, h. $A=\frac{1}{2}bh$

• Problem Solving Strategy for Geometry Applications Step 1. Read the problem and make sure you understand all the words and ideas. Draw the figure and label it with the given information. Step 2. Identify what you are looking for. Step 3. Name what you are looking for. Choose a variable to represent that quantity. Step 4. Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information. Step 5. Solve the equation using good algebra techniques. Step 6. Check the answer in the problem and make sure it makes sense. Step 7. Answer the question with a complete sentence.

• Properties of Circles $d=2r$ Circumference: $C=2\pi{}r$ or $C=\pi{}d$ Area: $A=\pi{r}^{2}$

• Volume and Surface Area of a Rectangular Solid $V=LWH$$S=2LH+2LW+2WH$

• Volume and Surface Area of a Cube $V={s}^{3}$$S=6{s}^{2}$

• Volume and Surface Area of a Sphere $V=\frac{4}{3}\pi{r}^{3}$$S=4\pi{r}^{2}$

• Volume and Surface Area of a Cylinder $V=\pi{r}^{2}h$$S=2\pi{r}^{2}+2\pi{}rh$

• Volume of a Cone For a cone with radius $r$ and height $h$: Volume: $V=\frac{1}{3}\pi{r}^{2}h$

• Distance, Rate, and Time $d=rt$

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Review Exercises

Approach Word Problems with a Positive Attitude

In the following exercises, solve.

Use a Problem Solving Strategy for Word Problems

In the following exercises, solve using the problem-solving strategy for word problems. Remember to write a complete sentence to answer each question.

Solve Number Problems

In the following exercises, solve each number word problem.

Solve Coin Word Problems

In the following exercises, solve each coin word problem.

Solve Ticket and Stamp Word Problems

In the following exercises, solve each ticket or stamp word problem.

Use Properties of Angles

In the following exercises, solve using properties of angles.

Use Properties of Triangles

In the following exercises, solve using properties of triangles.

In the following exercises, $\Delta{}ABC$ is similar to $\Delta{}XYZ.$ Find the length of the indicated side.

Use the Pythagorean Theorem

In the following exercises, use the Pythagorean Theorem to find the length of the missing side. Round to the nearest tenth, if necessary.

In the following exercises, solve. Approximate to the nearest tenth, if necessary.

Understand Linear, Square, Cubic Measure

In the following exercises, would you measure each item using linear, square, or cubic measure?

In the following exercises, find

1. ⓐ the perimeter

2. ⓑ the area of each figure

Use Properties of Rectangles

In the following exercises, find the ⓐ perimeter ⓑ area of each rectangle

In the following exercises, solve.

Use Properties of Triangles

In the following exercises, solve using the properties of triangles.

Use Properties of Trapezoids

In the following exercises, solve using the properties of trapezoids.

Use Properties of Circles

In the following exercises, solve using the properties of circles. Round answers to the nearest hundredth.

Find the Area of Irregular Figures

In the following exercises, find the area of each shaded region.

Find Volume and Surface Area of Rectangular Solids

In the following exercises, find the

1. ⓐ volume

2. ⓑ surface area of the rectangular solid

Find Volume and Surface Area of Spheres

In the following exercises, find the

1. ⓐ volume

2. ⓑ surface area of the sphere.

Find Volume and Surface Area of Cylinders

In the following exercises, find the

1. ⓐ volume

2. ⓑ surface area of the cylinder

Find Volume of Cones

In the following exercises, find the volume of the cone.

Use the Distance, Rate, and Time Formula

In the following exercises, solve using the formula for distance, rate, and time.

Solve a Formula for a Specific Variable

In the following exercises, use the formula $d=rt.$

In the following exercises, use the formula $A=\frac{1}{2}bh.$

In the following exercises, use the formula $I=Prt.$

In the following exercises, solve.

##

Practice Test

Key Terms

angle

An angle is formed by two rays that share a common endpoint. Each ray is called a side of the angle.

area

The area is a measure of the surface covered by a figure.

complementary angles

cone

A cone is a solid figure with one circular base and a vertex.

cube

A cube is a rectangular solid whose length, width, and height are equal.

cylinder

A cylinder is a solid figure with two parallel circles of the same size at the top and bottom.

equilateral triangle

A triangle with all three sides of equal length is called an equilateral triangle.

hypotenuse

The side of the triangle opposite the 90° angle is called the hypotenuse.

irregular figure

An irregular figure is a figure that is not a standard geometric shape. Its area cannot be calculated using any of the standard area formulas.

isosceles triangle

A triangle with two sides of equal length is called an isosceles triangle.

legs of a right triangle

The sides of a right triangle adjacent to the right angle are called the legs.

perimeter

The perimeter is a measure of the distance around a figure.

rectangle

A rectangle is a geometric figure that has four sides and four right angles.

right triangle

similar figures

In geometry, if two figures have exactly the same shape but different sizes, we say they are similar figures.

supplementary angles

trapezoid

A trapezoid is four-sided figure, a quadrilateral, with two sides that are parallel and two sides that are not.

triangle

A triangle is a geometric figure with three sides and three angles.

vertex of an angle

When two rays meet to form an angle, the common endpoint is called the vertex of the angle.

##

Key Concepts

• Problem Solving StrategyStep 1. Read the word problem. Make sure you understand all the words and ideas. You may need to read the problem two or more times. If there are words you don't understand, look them up in a dictionary or on the internet. Step 2. Identify what you are looking for. Step 3. Name what you are looking for. Choose a variable to represent that quantity. Step 4. Translate into an equation. It may be helpful to first restate the problem in one sentence before translating. Step 5. Solve the equation using good algebra techniques. Step 6. Check the answer in the problem. Make sure it makes sense. Step 7. Answer the question with a complete sentence.

• Finding the Total Value for Coins of the Same Type For coins of the same type, the total value can be found as follows: $\begin{array}{l} \mathrm{number}\cdot\mathrm{value}=\mathrm{total value} \\ \end{array}$where number is the number of coins, value is the value of each coin, and total value is the total value of all the coins.

• Solve a Coin Word ProblemStep 1. Read the problem. Make sure you understand all the words and ideas, and create a table to organize the information. Step 2. Identify what you are looking for. Step 3. Name what you are looking for. Choose a variable to represent that quantity.Use variable expressions to represent the number of each type of coin and write them in the table. Multiply the number times the value to get the total value of each type of coin. Step 4. Translate into an equation. Write the equation by adding the total values of all the types of coins. Step 5. Solve the equation using good algebra techniques. Step 6. Check the answer in the problem and make sure it makes sense. Step 7. Answer the question with a complete sentence.

• Type $\mathrm{Number}$$\mathrm{Value (\textdollar )}$$\mathrm{Total Value (\$)}$

• Supplementary and Complementary AnglesIf the sum of the measures of two angles is 180°, then the angles are supplementary. If $∠A$ and $∠B$ are supplementary, then $m∠A+m∠B=180$. If the sum of the measures of two angles is 90°, then the angles are complementary. If $∠A$ and $∠B$ are complementary, then $m∠A+m∠B=90$.

• Solve Geometry ApplicationsStep 1. Read the problem and make sure you understand all the words and ideas. Draw a figure and label it with the given information. Step 2. Identify what you are looking for. Step 3. Name what you are looking for and choose a variable to represent it. Step 4. Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information. Step 5. Solve the equation using good algebra techniques. Step 6. Check the answer in the problem and make sure it makes sense. Step 7. Answer the question with a complete sentence.

• Sum of the Measures of the Angles of a TriangleFor any $\Delta{}ABC,$ the sum of the measures is 180° $m∠A+m∠B+m∠C=180$

• Right Triangle A right triangle is a triangle that has one 90° angle, which is often marked with a $\measuredangle$symbol.

• Properties of Similar TrianglesIf two triangles are similar, then their corresponding angle measures are equal and their corresponding side lengths have the same ratio.

• Properties of Rectangles Rectangles have four sides and four right (90°) angles. The lengths of opposite sides are equal. The perimeter, $P$, of a rectangle is the sum of twice the length and twice the width. $P=2L+2W$ The area, $A$, of a rectangle is the length times the width. $A=L\cdot{}W$

• Triangle Properties For any triangle $\Delta{}ABC$, the sum of the measures of the angles is 180°. $m\mathrm{∠}A+m\mathrm{∠}B+m\mathrm{∠}C=180^{\circ}$ The perimeter of a triangle is the sum of the lengths of the sides. $P=a+b+c$ The area of a triangle is one-half the base, b, times the height, h. $A=\frac{1}{2}bh$

• Problem Solving Strategy for Geometry Applications Step 1. Read the problem and make sure you understand all the words and ideas. Draw the figure and label it with the given information. Step 2. Identify what you are looking for. Step 3. Name what you are looking for. Choose a variable to represent that quantity. Step 4. Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information. Step 5. Solve the equation using good algebra techniques. Step 6. Check the answer in the problem and make sure it makes sense. Step 7. Answer the question with a complete sentence.

• Properties of Circles $d=2r$ Circumference: $C=2\pi{}r$ or $C=\pi{}d$ Area: $A=\pi{r}^{2}$

• Volume and Surface Area of a Rectangular Solid $V=LWH$$S=2LH+2LW+2WH$

• Volume and Surface Area of a Cube $V={s}^{3}$$S=6{s}^{2}$

• Volume and Surface Area of a Sphere $V=\frac{4}{3}\pi{r}^{3}$$S=4\pi{r}^{2}$

• Volume and Surface Area of a Cylinder $V=\pi{r}^{2}h$$S=2\pi{r}^{2}+2\pi{}rh$

• Volume of a Cone For a cone with radius $r$ and height $h$: Volume: $V=\frac{1}{3}\pi{r}^{2}h$

• Distance, Rate, and Time $d=rt$

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Review Exercises

Approach Word Problems with a Positive Attitude

In the following exercises, solve.

Use a Problem Solving Strategy for Word Problems

In the following exercises, solve using the problem-solving strategy for word problems. Remember to write a complete sentence to answer each question.

Solve Number Problems

In the following exercises, solve each number word problem.

Solve Coin Word Problems

In the following exercises, solve each coin word problem.

Solve Ticket and Stamp Word Problems

In the following exercises, solve each ticket or stamp word problem.

Use Properties of Angles

In the following exercises, solve using properties of angles.

Use Properties of Triangles

In the following exercises, solve using properties of triangles.

In the following exercises, $\Delta{}ABC$ is similar to $\Delta{}XYZ.$ Find the length of the indicated side.

Use the Pythagorean Theorem

In the following exercises, use the Pythagorean Theorem to find the length of the missing side. Round to the nearest tenth, if necessary.

In the following exercises, solve. Approximate to the nearest tenth, if necessary.

Understand Linear, Square, Cubic Measure

In the following exercises, would you measure each item using linear, square, or cubic measure?

In the following exercises, find

1. ⓐ the perimeter

2. ⓑ the area of each figure

Use Properties of Rectangles

In the following exercises, find the ⓐ perimeter ⓑ area of each rectangle

In the following exercises, solve.

Use Properties of Triangles

In the following exercises, solve using the properties of triangles.

Use Properties of Trapezoids

In the following exercises, solve using the properties of trapezoids.

Use Properties of Circles

In the following exercises, solve using the properties of circles. Round answers to the nearest hundredth.

Find the Area of Irregular Figures

In the following exercises, find the area of each shaded region.

Find Volume and Surface Area of Rectangular Solids

In the following exercises, find the

1. ⓐ volume

2. ⓑ surface area of the rectangular solid

Find Volume and Surface Area of Spheres

In the following exercises, find the

1. ⓐ volume

2. ⓑ surface area of the sphere.

Find Volume and Surface Area of Cylinders

In the following exercises, find the

1. ⓐ volume

2. ⓑ surface area of the cylinder

Find Volume of Cones

In the following exercises, find the volume of the cone.

Use the Distance, Rate, and Time Formula

In the following exercises, solve using the formula for distance, rate, and time.

Solve a Formula for a Specific Variable

In the following exercises, use the formula $d=rt.$

In the following exercises, use the formula $A=\frac{1}{2}bh.$

In the following exercises, use the formula $I=Prt.$

In the following exercises, solve.

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Practice Test

1. Francie has $\text{\textdollar 4.35}$ in dimes and quarters. The number of dimes is $5$ more than the number of quarters. How many of each coin does she have?

2. Paulette has $\text{\textdollar 140}$ in $\text{\textdollar 5}$ and $\text{\textdollar 10}$ bills. The number of $\text{\textdollar 10}$ bills is one less than twice the number of $\text{\textdollar 5}$ bills. How many of each does she have?

3. A church luncheon made $\text{\textdollar 842}.$ Adult tickets cost $\text{\textdollar 10}$ each and children’s tickets cost $\text{\textdollar 6}$ each. The number of children was $12$ more than twice the number of adults. How many of each ticket were sold?

4. Ana spent $\text{\textdollar 4.06}$ buying stamps. The number of $\text{\textdollar 0.41}$ stamps she bought was $5$ more than the number of $\text{\textdollar 0.26}$ stamps. How many of each did she buy?

1. What is the supplement of a $\mathrm{48^{\circ}}$ angle?

2. Two angles are complementary. The smaller angle is $\mathrm{24^{\circ}}$ less than the larger angle. Find the measures of both angles.

3. The measures of two angles of a triangle are $22$ and $85$ degrees. Find the measure of the third angle.

4. One angle of a triangle is $\mathrm{30^{\circ}}$ more than the smallest angle. The largest angle is the sum of the other angles. Find the measures of all three angles.

1. Sergio needs to attach a wire to hold the antenna to the roof of his house, as shown in the figure. The antenna is $8$ feet tall and Sergio has $10$ feet of wire. How far from the base of the antenna can he attach the wire?

2. amount of sand in a sandbag

3. size of a patio

4. The length of a rectangle is $42$ meters and the width is $28$ meters.

1. A sidewalk in front of Kathy’s house is in the shape of a rectangle $4$ feet wide by $45$ feet long.

2. Find the length of a rectangle with perimeter of $220$ centimeters and width of $85$ centimeters.

3. The area of a rectangle is $2356$ square meters. The length is $38$ meters. What is the width?

4. The length of a rectangle is $12$ centimeters more than the width. The perimeter is $74$ centimeters. Find the length and the width.

1. Find the area of a triangle with base $18$ inches and height $15$ inches.

2. A triangular road sign has base $30$ inches and height $40$ inches. What is its area?

3. A tile in the shape of an isosceles triangle has a base of $6$ inches. If the perimeter is $20$ inches, find the length of each of the other sides.

4. The perimeter of a triangle is $59$ feet. One side of the triangle is $3$ feet longer than the shortest side. The third side is $5$ feet longer than the shortest side. Find the length of each side.

1. The height of a trapezoid is $8$ feet and the bases are $11$ and $14$ feet. What is the area?

2. Find the area of the trapezoid with height $25$ meters and bases $32.5$ and $21.5$ meters.

3. A circular mosaic has radius $3$ meters. Find the

1. ⓐ circumference

2. ⓑ area of the mosaic

4. Find the diameter of a circle with circumference $150.72$ inches.

1. a rectangular solid with length $14$ centimeters, width $4.5$ centimeters, and height $10$ centimeters

2. a cube of tofu with sides $2.5$ inches

3. a sphere with radius $4$ yards

4. a baseball with radius $1.45$ inches

1. a cylinder with radius $2$ yards and height $6$ yards

2. a juice can with diameter $8$ centimeters and height $15$ centimeters

3. a cone with height $5$ meters and radius $1$ meter

4. a cone-shaped water cup with diameter $2.6$ inches and height $2.6$ inches

1. A plane flew $4$ hours at $380$ miles per hour. What distance was covered?

2. Jack is driving from Bangor to Portland at a rate of $68$ miles per hour. The distance is $107$ miles. To the nearest tenth of an hour, how long will the trip take?

3. Solve for $t:$

1. ⓐ when $d=403$ and $r=65$

2. ⓑ in general

4. Solve for $b:$

1. ⓐ when $A=416$ and $h=32$

2. ⓑ in general

1. Solve for the principal, $P,$ for:

1. ⓐ $\,I=\text{\textdollar 720}$, $r=\mathrm{4\%}$, $t=3\,\mathrm{years}$

2. ⓑ in general

2. Solve the formula $6x+5y=20$ for $y:$

1. ⓐ when $x=0$

2. ⓑ in general

3. Solve $a+b=90$ for $a.$

4. Solve the formula $4x+y=17$ for $y.$

1. Solve the formula $P=2L+2W$ for $W.$

2. The sum of $13$ and twice a number is $-19.$ Find the number.

3. Bonita has $\text{\textdollar 2.95}$ in dimes and quarters in her pocket. If she has $5$ more dimes than quarters, how many of each coin does she have?

4. Find the complement of a $\mathrm{52^{\circ}}$ angle.

1. A baseball diamond is shaped like a square with sides $90$ feet long. How far is it from home plate to second base, as shown?

2. A triangular poster has base $80$ centimeters and height $55$ centimeters. Find the area of the poster.

3. A circular pool has diameter $90$ inches. What is its circumference? Round to the nearest tenth.

4. A traffic cone has height $75$ centimeters. The radius of the base is $20$ centimeters. Find the volume of the cone. Round to the nearest tenth.

1. How has your attitude towards solving word problems changed as a result of working through this chapter? Explain.

2. Three-fourths of the people at a concert are children. If there are $87$ children, what is the total number of people at the concert?

3. Reza was very sick and lost $\mathrm{15\%}$ of his original weight. He lost $27$ pounds. What was his original weight?

4. One number is nine less than another. Their sum is twenty-seven. Find the numbers.

1. The Catalina Express takes $1\frac{1}{2}$ hours to travel from Long Beach to Catalina Island, a distance of $22$ miles. To the nearest tenth, what is the speed of the boat?

2. Solve the formula $A=\frac{1}{2}bh$ for $h:$

1. ⓐ when $A=1716$ and $b=66$

2. ⓑ in general