Title: Unit 7 Quiz
Course: Technical Math for Industry

Q: Identify all rays and lines in the picture below.
- Rays:

Lines: (Incorrect. The choices named are all correct, but not all the rays in this picture have been included. You may not have realized that there can be two different rays through the same two points.)
- Rays:

Lines: (Incorrect. The choices named are all correct, but not all the rays in this picture have been included. You may not have realized that any line also contains rays.)
- Rays:

Lines: (Incorrect. First, you may not have realized that any line also contains rays. Second, a line continues infinitely in both directions. This is indicated by arrows on both ends. If there is an arrow on only one end, this is not a line.)
- Rays:

Lines: (Correct. A line continues infinitely in both directions. This is indicated by arrows on both ends, so is the only line. A ray continues infinitely in one direction. This is indicated by an arrow on the end. For each of the four arrows in the picture there is a ray: , (points to the left), (points to the right), and)
A: Rays:

Lines:

Q: What is the measurement of the angle shown below?
A: (a choice with no text)
If wrong: Incorrect. The angle shown is acute, so its measure will be less than . The protractor is aligned with the ray on the left at . Because is on the outside, the outside number to which the other ray is pointing will be the angle measure.
If wrong: Incorrect. Perhaps you overlooked the horizontal ray and found the angle that the other ray would make with a vertical line. The protractor is aligned with the ray on the left at . Because is on the outside, the outside number to which the other ray is pointing will be the angle measure.
If right: Correct. The angle shown is acute, so its measure will be less than . The protractor is aligned with the ray on the left at . Because is on the outside, the outside number to which the other ray is pointing will be the angle measure. This is .
If wrong: Incorrect. The angle shown is acute, so its measure will be less than . The protractor is aligned with the ray on the left at . Because is on the outside, the outside number to which the other ray is pointing will be the angle measure.

Q: Which two statements are both true?
- and (Incorrect. Although one of the statements is true, the other one is not. Lines are known to be perpendicular if there is a right angle mark at their intersection.)
- and (Correct. The first pair of lines is perpendicular, as indicated by the right angle mark at their intersection. The second pair of lines is parallel because they are perpendicular to the same line.)
- and (Incorrect. Although one of the statements is true, the other one is not. Lines are known to be parallel if they have small double arrow symbols or if they are perpendicular to the same line. Lines are known to be perpendicular if they have a right angle mark at their intersection.)
- and (Incorrect. Although one of the statements is true, the other one is not. Lines are known to be parallel if they have small double arrow symbols or if they are perpendicular to the same line.)
A: and

Q: Find the measure of .
A: (a choice with no text)
If wrong: Incorrect. Perhaps you thought that was the same size as . The four angles along the straight angle have measures that total . Find the sum of the three given angles and then subtract that sum from 180.
If right: Correct. Because is a straight angle, it measures . The four angles along this straight angle have measures that total . So:
If wrong: Incorrect. You may have realized that is a straight angle, and therefore the sum of the four angle measures is . But perhaps you added incorrectly.
If wrong: Incorrect. You may have correctly found the sum of the three angle measures, but then subtracted that sum from 180 incorrectly.

Q: Look at and in the image below.

Which of the following is the best description for this pair of angles?
- complementary (Incorrect. Perhaps you confused the definition of supplementary angles with the definition of complementary angles.)
- acute (Incorrect. One of the angles is acute. However, the word ''acute'' does not describe the pair of angles.)
- straight (Incorrect. The pair of angles together form a single angle. That single angle is a straight angle. However, the word ''straight'' does not describe the pair of angles.)
- supplementary (Correct. The two given angles are on a straight line, so their measurements will add up to . This makes them supplementary angles.)
A: supplementary

Q: How would the following triangle be classified?
- scalene acute (Incorrect. The two notches indicate that the left and right sides are congruent. Also, the top angle is greater than .)
- isosceles acute (Incorrect. Because two sides are congruent, the triangle is isosceles. However, the top angle is greater than .)
- scalene obtuse (Incorrect. Because the top angle is greater than , the triangle is obtuse. However, the two notches indicate that the left and right sides are congruent.)
- isosceles obtuse (Correct. Because two sides are congruent, the triangle is isosceles. Because the top angle is greater than , the triangle is obtuse.)
A: isosceles obtuse

Q: Which of the following pairs show(s) two congruent triangles?
- A, B, and C (Incorrect. You may have confused the definitions of congruent and similar. All three pairs show similar triangles because, as indicated by the markings, they have the same angles (the same shape). However, for two triangles to be congruent, they must also have the same size.)
- B only (Correct. All sides and angles in the first triangle in B are congruent to all sides and angles in the second triangle in B, as indicated by the markings. Therefore, the triangles in B are congruent. The first and third pairs of triangles are similar, as indicated by the fact that they have the same angles.)
- B and C only (Incorrect. Two triangles are congruent when they have the same size (same side lengths) and the same shape (same angles). This will be shown by markings or by actual numbers. If sides, for example, seem to have about the same length, but there are no markings or lengths to show this, then you cannot assume that they have the same length.)
- A and C only (Incorrect. You may have confused the definitions of congruent and similar. These pairs show similar triangles because, as indicated by the markings, they have the same angles (the same shape). However, for two triangles to be congruent, they must also have the same size.)
A: B only

Q: The drawing below shows that the two triangles are similar.

Write the corresponding sides as equal ratios.
A: (a choice with no text)
If wrong: Incorrect. The sides of similar triangles are proportional, so the ratios of corresponding sides are equal. Corresponding sides will be located opposite corresponding (congruent) angles. For example, the measure of equals the measure of , so NR does correspond to JL, which is the ratio in the middle. However, the other ratios should also be of lengths in the first triangle to lengths in the second triangle, which they are not.
If wrong: Incorrect. The sides of similar triangles are proportional, so the ratios of corresponding sides are equal. Corresponding sides will be located opposite corresponding (congruent) angles. For example, the measure of equals the measure of , so MN corresponds to BJ, not to BL.
If right: Correct. The sides of similar triangles are proportional, so the ratios of corresponding sides are equal. Corresponding sides will be located opposite corresponding (congruent) angles. For example, the measure of equals the measure of , so MN corresponds to BJ, and these two lengths are used in the first ratio.
If wrong: Incorrect. The sides of similar triangles are proportional, so the ratios of corresponding sides are equal. Corresponding sides will be located opposite corresponding (congruent) angles. For example, the measure of equals the measure of , so MN corresponds to BJ, and these two lengths are used in the first ratio. The second ratio is also correct, but the last one should also be of a length in the first triangle to a length in the second triangle, which it is not.

Q: In the triangles below, and .

What is the length of ?
- 12 (Incorrect. You may have set up the proportion incorrectly. corresponds to , and corresponds to . Now set up the correct proportion, cross-multiply, and solve.)
- 6 (Incorrect. You may have thought that because 8 is 2 more than 6, PB would be 2 more than 4. However, in similar triangles, side lengths are proportional. corresponds to , and corresponds to . Now set up the correct proportion and solve.)
- 3 (Incorrect. It appears that you set up the proportion incorrectly. corresponds to , and corresponds to . Now set up the correct proportion, cross-multiply, and solve.)
A: (a choice with no text)
If right: Correct. Let x represent the unknown side. Set up a proportion of corresponding sides:

Q: A flashlight is projecting a triangle onto a wall, as shown below.

The original triangle and its projection are similar. What is the missing length n on the projection?
- 12.6 (Incorrect. You mixed up the corresponding sides when setting up the proportion. In this picture, the bases 21 and n are corresponding sides. The vertical sides 15 and 25 are corresponding as well. Set up the proportion and solve.)
- 31 (Incorrect. You may have added 10 to 21, because 25 is 10 more than 15. This is not the correct way to find a missing side in a similar triangle. In this picture, the bases 21 and n are corresponding sides. The vertical sides 15 and 25 are corresponding as well. Set up the proportion and solve.)
- 17.9 (Incorrect. You mixed up the corresponding sides when setting up the proportion. In this picture, the bases 21 and n are corresponding sides. The vertical sides 15 and 25 are corresponding as well. Set up the proportion and solve.)
- 35 (Correct.)
A: 35

Q: Find the length of the missing leg in the following right triangle.
- 42.0 (Incorrect. You probably used the number 41 as the length of a leg instead of the hypotenuse. A proper use of the Pythagorean Theorem will give you . Solve for b.)
- 32 (Incorrect. It appears that you just subtracted the leg length from the hypotenuse. You need to use the Pythagorean Theorem: . Substitute the values and , and then solve for b.)
- 1,600 (Incorrect. You probably correctly applied the Pythagorean Theorem, but then forgot to take the square root. You need to solve for b.)
- 40 (Correct.)
A: 40

Q: Hermina cut a 10'' by 15'' piece of cardboard down the diagonal.

What is the length c of the cut, in inches?
- 25 (Incorrect. It appears that you just added the 10 and the 15. The cardboard is a rectangle. The diagonal and the sides form a right triangle. Use the Pythagorean Theorem: . Solve for c.)
A: (a choice with no text)
If wrong: Incorrect. You may have used the Pythagorean Theorem with 10 as a leg and the 15 as the hypotenuse. However, the diagonal is the hypotenuse. This gives you the equation . Solve for c.
If wrong: Incorrect. Perhaps you doubled the 10 and 15 instead of squaring them. The sides are legs of a right triangle, and the diagonal is the hypotenuse. Use the Pythagorean Theorem: . Solve for c.
If right: Correct. The cardboard is a rectangle. The diagonal and the sides form a right triangle. Use the Pythagorean Theorem:

Q: Which of the following statements is true?
- A trapezoid must have congruent sides. (Incorrect. A trapezoid is a quadrilateral with exactly one pair of parallel sides. These parallel sides are not congruent. The sides that are not parallel could be congruent, but they do not have to be.)
- All rectangles are squares. (Incorrect. A rectangle is defined as a quadrilateral with four right angles. Adjacent sides may have different lengths. A square is defined as a rectangle with four congruent sides. Because not every rectangle has four congruent sides, not all rectangles are squares.)
- All rectangles are parallelograms. (Correct. A rectangle is defined as a quadrilateral with four right angles. If the quadrilateral has four right angles, opposite sides must be parallel. So it is also a parallelogram.)
- Every angle in a quadrilateral must be less than . (Incorrect. Most quadrilaterals that you will see have angles less than , but it is possible for a quadrilateral to have one angle that is greater than .)
A: All rectangles are parallelograms.

Q: Find the perimeter, in feet, of the parallelogram.
- 56 (Correct. Opposite sides of a parallelogram have the same lengths, so use the formula to find the perimeter.)
- 28 (Incorrect. You added only the two given side lengths of the parallelogram, but you need to add all four side lengths.)
- 47 (Incorrect. You added the three measurements labeled on the figure, instead of finding the sum of the lengths of the four sides of the parallelogram.)
- 54 (Incorrect. You used the height of the parallelogram instead of the length when calculating the perimeter.)
A: 56

Q: A square tile has a side length of 12 cm. What is the area of the tile?
- 72 square centimeters (Incorrect. You seem to have multiplied the area of the tile by , so you may have been thinking about the area of a triangle. The formula for the area of a square is simply:)
- 24 square centimeters (Incorrect. You multiplied the side length of the tile by 2, instead of squaring the side length to find the area.)
- 48 square centimeters (Incorrect. It appears that you found the perimeter of the tile, instead of the area.)
- 144 square centimeters (Correct. The formula for the area of a square is: Substitute the value : The area of the square tile is 144 square centimeters.)
A: 144 square centimeters

Q: Find the area of the figure below, formed from a triangle and a rectangle, in square millimeters.
- 252 (Incorrect. You may have forgotten the in the formula for the area of a triangle, and also mistakenly multiplied the area of the rectangle by , thinking of the formula for a triangle or a trapezoid.)
- 156 (Incorrect. You may have mistakenly multiplied the area of the rectangle by , thinking of the formula for a triangle or a trapezoid.)
- 312 (Incorrect. You may have forgotten the in the formula for the area of a triangle.)
- 216 (Correct. Add the areas of the triangle and the rectangle. Area of the triangle: Area of the rectangle: The total area of the figure is square millimeters.)
A: 216

Q: Which letter on the diagram below represents a diameter of the circle?
- B (Correct. A diameter of a circle is a line segment that passes through the center of a circle and has its endpoints on the circle itself. The line segment labeled with the letter B is a diameter.)
- A (Incorrect. The letter A represents the center of the circle. A diameter of a circle is a line segment that passes through the center of a circle and has its endpoints on the circle itself.)
- D (Incorrect. The letter D represents the circumference of the circle or the circle itself. A diameter of a circle is a line segment that passes through the center of a circle and has its endpoints on the circle itself.)
- C (Incorrect. The letter C represents a radius of the circle. A diameter of a circle is a line segment that passes through the center of a circle and has its endpoints on the circle itself.)
A: B

Q: Find the approximate circumference of a circle with a diameter of 100 yards.
- 31 yards (Incorrect. You may have multiplied wrong when evaluating the formula for .)
- 628 yards (Incorrect. You may have thought that the circumference equals . The formula for the circumference of a circle is or . In this circle, .)
- 314 yards (Correct. The circumference of a circle is given by the formula , with . In this circle, , so: The approximate circumference of the circle is 314 yards.)
- 157 yards (Incorrect. You multiplied by the radius of the circle, instead of multiplying it by the diameter.)
A: 314 yards

Q: The clock face in a famous clock tower has a radius of about 4 meters. What is the area of the clock face to the nearest square meter? Use 3.14 as an approximation for .
- 50 square meters (Correct. Use in the formula to find the area: Rounded to the nearest square meter, the area of the clock face is 50 square meters.)
- 13 square meters (Incorrect. You may have thought that the diameter of the clock face was 4 meters and divided the value by 2 before using the area formula. However, the radius of the clock face is 4 meters.)
- 25 square meters (Incorrect. You used the formula for the circumference of a circle instead of the area.)
- 201 square meters (Incorrect. You used the square of the diameter in the area formula instead of the square of the radius.)
A: 50 square meters

Q: Find the approximate area of the shaded region below, consisting of a right triangle with a circle cut out of it. Use 3.14 as an approximation for .
- 314 square meters (Incorrect. You found the area of the circle. You need to subtract this from the area of the triangle to find the area of the shaded region.)
- 3,286 square meters (Incorrect. You probably forgot to use in the formula for the area of a triangle, and only multiplied the base by the height.)
- 544 square meters (Incorrect. You probably found the area of the circle using 20 as the radius instead of 10. Find the approximate area of the circle with and then subtract this from the area of the square.)
- 1,486 square meters (Correct. Use the radius of the circle, , to find the area of the circle: square meters. The area of the triangle is: The area of the shaded region equals the area of the triangle minus the area of the circle. This is approximately square meters.)
A: 1,486 square meters

Q: A three-dimensional figure has the following properties:

It has a circular base.

It has a second, circular base parallel to the first, with a round smooth face between them.

What kind of solid is it?
- sphere (Incorrect. A sphere has no base. Because the solid has a circular base, it must be either a cylinder or a cone. The second property mentioned in the problem lets you determine which of these solids it is.)
- cone (Incorrect. A cone does have a circular base, but it has only one. The object described has two circular bases.)
- rectangular prism (Incorrect. A rectangular prism has a rectangular base. Because the solid has a circular base, it must be either a cylinder or a cone. The second property mentioned in the problem lets you determine which of these solids it is.)
- cylinder (Correct. A circular base indicates that the figure is either a cylinder or a cone. Because there are two parallel bases with a smooth face between them, the figure is a cylinder.)
A: cylinder

Q: Find the volume of a sphere with a diameter of 6 centimeters.
- 36 cubic centimeters (Correct. The volume of a sphere is: The radius of the sphere is half the diameter, or centimeters. Substitute into the formula: cubic centimeters)
- 288 cubic centimeters (Incorrect. You may have used the diameter in the formula for the volume of a sphere. Note that is the correct formula for the volume of a sphere.)
- 9 cubic centimeters (Incorrect. You may have been thinking of the formula for the volume of a cone or pyramid, and used as the constant factor instead of . Note that is the correct formula for the volume of a sphere.)
- 12 cubic centimeters (Incorrect. You may have only squared the radius in the formula for the volume of a sphere. Note that is the correct formula for the volume of a sphere.)
A: 36 cubic centimeters

Q: The three-dimensional figure below is a cylinder with a hole in the shape of a rectangular prism going through the center of it. The radius is 10 yards. Find the volume of the solid in cubic yards, rounded to the nearest ten. Use 3.14 for .
- 725 (Incorrect. You probably mixed up the radius and the height when substituting into to find the volume of the cylinder.)
- 1,510 (Correct. First find the volume of the cylinder: Then find the volume of the hole shaped like a rectangular prism: Subtract the volume of the hole from the volume of the cylinder: The volume of the solid is 1,510 cubic yards.)
- 1,570 (Incorrect. You found the volume of the cylinder, but did not subtract the volume of the hole that is going through the center of it.)
- 1,630 (Incorrect. You probably correctly found the volume of the cylinder and the volume of the hole shaped like a rectangular prism. However, you added the two volumes instead of subtracting the volume of the hole from the solid.)
A: 1,510