Integrated Mathematics (Im)2 Unit 5 Exam

1. If a quadrilateral is a rectangle then its diagonals are congruent.

True

False

If a quadrilateral is a rectangle, it means that it has four right angles. In a rectangle, the diagonals are always congruent, meaning that they have the same length. This is because the diagonals of a rectangle bisect each other and form two congruent right triangles. Therefore, if a quadrilateral is a rectangle, its diagonals are congruent.

Explanation

If a quadrilateral is a rectangle, it means that it has four right angles. In a rectangle, the diagonals are always congruent, meaning that they have the same length. This is because the diagonals of a rectangle bisect each other and form two congruent right triangles. Therefore, if a quadrilateral is a rectangle, its diagonals are congruent.

2. In triangle ABC , point D is the midpoint of AC and E is the midpoint of AB. If BC = 22, what is the length of DE?

5

10

11

2

In triangle ABC, point D is the midpoint of AC and E is the midpoint of AB. Since D is the midpoint of AC, AD and DC are equal in length. Similarly, since E is the midpoint of AB, AE and EB are equal in length. Therefore, triangle ADE is a right triangle with DE as the hypotenuse. Using the Pythagorean theorem, we can find the length of DE. Since AD = DC = 11 (half of 22), DE can be calculated as √(11^2 + 11^2) = √(121 + 121) = √242 ≈ 11.

Explanation

In triangle ABC, point D is the midpoint of AC and E is the midpoint of AB. Since D is the midpoint of AC, AD and DC are equal in length. Similarly, since E is the midpoint of AB, AE and EB are equal in length. Therefore, triangle ADE is a right triangle with DE as the hypotenuse. Using the Pythagorean theorem, we can find the length of DE. Since AD = DC = 11 (half of 22), DE can be calculated as √(11^2 + 11^2) = √(121 + 121) = √242 ≈ 11.

3. In the kite below which pair of angles is congruent?

A and D

B and D

A and C

B and C

In the given kite, angles A and C are congruent. This is because in a kite, the angles formed by the intersecting diagonals are congruent. Therefore, angle A and angle C are equal in measure.

Explanation

In the given kite, angles A and C are congruent. This is because in a kite, the angles formed by the intersecting diagonals are congruent. Therefore, angle A and angle C are equal in measure.

4. What is the image of point A(3,4) after a dilation of factor 3?

A'(6, 7)

A'(1, 0.75)

A'(9, 12)

A'(0,1)

The image of point A(3,4) after a dilation of factor 3 can be found by multiplying the coordinates of point A by the dilation factor. In this case, when we multiply 3 by 3, we get 9, and when we multiply 4 by 3, we get 12. Therefore, the image of point A after the dilation is A'(9, 12).

Explanation

The image of point A(3,4) after a dilation of factor 3 can be found by multiplying the coordinates of point A by the dilation factor. In this case, when we multiply 3 by 3, we get 9, and when we multiply 4 by 3, we get 12. Therefore, the image of point A after the dilation is A'(9, 12).

5. "I am a quadrilateral, I have two pairs of consecutive congruent sides. Which of the following am NOT?"

Kite

Rhombus

Isosceles Trapezoid

Square

An isosceles trapezoid does not have two pairs of consecutive congruent sides. It has only one pair of congruent sides, which are the bases of the trapezoid. Therefore, it does not fit the given criteria and is the correct answer.

Explanation

An isosceles trapezoid does not have two pairs of consecutive congruent sides. It has only one pair of congruent sides, which are the bases of the trapezoid. Therefore, it does not fit the given criteria and is the correct answer.

6. When two lines are parallel, __________________________________.

They have equal slopes

Their slopes are negative reciprocals of each other

They are also perpendicular

The product of their slopes = -1

When two lines are parallel, they have equal slopes. This means that the rate of change of the lines, or the steepness, is the same for both lines. If the slopes were different, the lines would intersect at some point, indicating that they are not parallel. Therefore, when two lines are parallel, their slopes must be equal.

Explanation

When two lines are parallel, they have equal slopes. This means that the rate of change of the lines, or the steepness, is the same for both lines. If the slopes were different, the lines would intersect at some point, indicating that they are not parallel. Therefore, when two lines are parallel, their slopes must be equal.

7. Which of the following lines is parallel to: 9x + 3y = 22?

Y = -3x + 43

Y = -0.33x - 9

Y = 0.33x + 10

Y = 3x - 1

The given equation is in the form of "ax + by = c". In order for a line to be parallel to this equation, it must have the same slope. The slope of the given equation can be found by rearranging it in the form "y = mx + b", where m is the slope. In this case, the slope is -3. Therefore, the line y = -3x + 43 has the same slope and is parallel to the given equation.

Explanation

The given equation is in the form of "ax + by = c". In order for a line to be parallel to this equation, it must have the same slope. The slope of the given equation can be found by rearranging it in the form "y = mx + b", where m is the slope. In this case, the slope is -3. Therefore, the line y = -3x + 43 has the same slope and is parallel to the given equation.

8. The following translation is: (x,y) --> (-x,-y)

A reflection on x-axis

A reflection on y-axis

180 degree rotation

360 degree rotation

The given translation maps any point (x, y) to the point (-x, -y). This means that the x-coordinate is negated and the y-coordinate is negated. This transformation is equivalent to rotating the point 180 degrees around the origin. Therefore, the correct answer is 180 degree rotation.

Explanation

The given translation maps any point (x, y) to the point (-x, -y). This means that the x-coordinate is negated and the y-coordinate is negated. This transformation is equivalent to rotating the point 180 degrees around the origin. Therefore, the correct answer is 180 degree rotation.

9. All parallelograms are squares.

True

False

The statement "All parallelograms are squares" is false. While all squares are parallelograms, not all parallelograms are squares. A square is a special type of parallelogram where all sides are equal in length and all angles are right angles. However, a parallelogram can have unequal sides and angles that are not right angles. Therefore, the statement is incorrect.

Explanation

The statement "All parallelograms are squares" is false. While all squares are parallelograms, not all parallelograms are squares. A square is a special type of parallelogram where all sides are equal in length and all angles are right angles. However, a parallelogram can have unequal sides and angles that are not right angles. Therefore, the statement is incorrect.

10.

The best name for this quadrilateral is:________________________.

A quadrilateral with opposite sides that are parallel is known as a parallelogram. In this case, the answer "parallelogram" or "Parallelogram" is the best name for the given quadrilateral as it indicates that the opposite sides of the shape are parallel to each other.

Explanation

A quadrilateral with opposite sides that are parallel is known as a parallelogram. In this case, the answer "parallelogram" or "Parallelogram" is the best name for the given quadrilateral as it indicates that the opposite sides of the shape are parallel to each other.

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11. Some parallelograms are squares.

True

False

Parallelograms are a type of quadrilateral with opposite sides that are parallel. A square is a special type of parallelogram where all sides are equal in length and all angles are right angles. Since a square meets the definition of a parallelogram, it can be concluded that some parallelograms are indeed squares. Therefore, the statement "Some parallelograms are squares" is true.

Explanation

Parallelograms are a type of quadrilateral with opposite sides that are parallel. A square is a special type of parallelogram where all sides are equal in length and all angles are right angles. Since a square meets the definition of a parallelogram, it can be concluded that some parallelograms are indeed squares. Therefore, the statement "Some parallelograms are squares" is true.

12. The diagonals of all rhombi are perpendicular.

True

False

The statement is true because in a rhombus, the opposite sides are parallel and the opposite angles are equal. Therefore, the diagonals of a rhombus bisect each other and form four right angles at their intersection point, making them perpendicular to each other.

Explanation

The statement is true because in a rhombus, the opposite sides are parallel and the opposite angles are equal. Therefore, the diagonals of a rhombus bisect each other and form four right angles at their intersection point, making them perpendicular to each other.

13. If a quadrilateral is a rhombus then it is a square.

True

False

The statement is false because while all squares are rhombuses, not all rhombuses are squares. A rhombus is a quadrilateral with all sides of equal length, but its angles do not have to be right angles. On the other hand, a square is a special type of rhombus where all sides are equal in length and all angles are right angles. Therefore, a rhombus can be a square, but it is not always the case.

Explanation

The statement is false because while all squares are rhombuses, not all rhombuses are squares. A rhombus is a quadrilateral with all sides of equal length, but its angles do not have to be right angles. On the other hand, a square is a special type of rhombus where all sides are equal in length and all angles are right angles. Therefore, a rhombus can be a square, but it is not always the case.

14. Which quadrilateral has one pair of parallel sides?

Square

Trapezoid

Rectangle

Kite

A trapezoid is a quadrilateral that has one pair of parallel sides. In a trapezoid, the opposite sides are not parallel, but one pair of sides is parallel. This distinguishes it from other quadrilaterals like squares, rectangles, and kites, which do not have parallel sides. Therefore, the correct answer is trapezoid.

Explanation

A trapezoid is a quadrilateral that has one pair of parallel sides. In a trapezoid, the opposite sides are not parallel, but one pair of sides is parallel. This distinguishes it from other quadrilaterals like squares, rectangles, and kites, which do not have parallel sides. Therefore, the correct answer is trapezoid.

15. If line j has slope = 0/5 and k does not have a slope, then j is ___________ to k.

Parallel

Perpendicular

Congruent

Opposite

If line j has a slope of 0/5, it means that it is a horizontal line. On the other hand, if line k does not have a slope, it means that it is a vertical line. Perpendicular lines are lines that intersect at a right angle, which is the case when a horizontal line intersects with a vertical line. Therefore, line j is perpendicular to line k.

Explanation

If line j has a slope of 0/5, it means that it is a horizontal line. On the other hand, if line k does not have a slope, it means that it is a vertical line. Perpendicular lines are lines that intersect at a right angle, which is the case when a horizontal line intersects with a vertical line. Therefore, line j is perpendicular to line k.

16. Every square is a rhombus.

True

False

A square is a special type of rhombus where all four sides are equal in length and all four angles are right angles. Therefore, every square can be considered as a rhombus.

Explanation

A square is a special type of rhombus where all four sides are equal in length and all four angles are right angles. Therefore, every square can be considered as a rhombus.

17.

True

False

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Explanation

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18. Triangle ABCD has vertices at A(0,6), B(2,5), C(-3,4). Which vertex is closest to the origin?

A

B

C

Vertex C is closest to the origin because it has the smallest distance from the origin compared to the other vertices. The distance between two points in a coordinate plane can be calculated using the distance formula, which is the square root of the sum of the squares of the differences in the x-coordinates and the y-coordinates. By calculating the distance between the origin (0,0) and each vertex, it can be determined that the distance between the origin and vertex C is the smallest, making it the closest vertex to the origin.

Explanation

Vertex C is closest to the origin because it has the smallest distance from the origin compared to the other vertices. The distance between two points in a coordinate plane can be calculated using the distance formula, which is the square root of the sum of the squares of the differences in the x-coordinates and the y-coordinates. By calculating the distance between the origin (0,0) and each vertex, it can be determined that the distance between the origin and vertex C is the smallest, making it the closest vertex to the origin.

19. If a quadrilateral has perpendicular diagonals, it cannot be a:___________________

Square

Rhombus

Kite

Trapezoid

A trapezoid is a quadrilateral that has only one pair of parallel sides. Perpendicular diagonals would imply that the opposite sides are equal in length and the trapezoid would then become a parallelogram. Therefore, a trapezoid cannot have perpendicular diagonals.

Explanation

A trapezoid is a quadrilateral that has only one pair of parallel sides. Perpendicular diagonals would imply that the opposite sides are equal in length and the trapezoid would then become a parallelogram. Therefore, a trapezoid cannot have perpendicular diagonals.

20. When two lines are __________________, they intersect and form 90 degree angles.

When two lines are perpendicular, it means that they intersect with each other at a right angle, forming 90-degree angles.

Explanation

When two lines are perpendicular, it means that they intersect with each other at a right angle, forming 90-degree angles.

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21. Find the missing coordinate of a parallelogram with coordinates at (0,0), (8,0), (x, 7) and (2,7)

The missing coordinate of the parallelogram can be found by observing that opposite sides of a parallelogram are equal in length. In this case, the length of the side connecting (0,0) and (8,0) is 8 units. Therefore, the length of the side connecting (x,7) and (2,7) must also be 8 units. Since the y-coordinate is the same for both points, the difference in x-coordinates must be 8. Therefore, x = 10.

Explanation

The missing coordinate of the parallelogram can be found by observing that opposite sides of a parallelogram are equal in length. In this case, the length of the side connecting (0,0) and (8,0) is 8 units. Therefore, the length of the side connecting (x,7) and (2,7) must also be 8 units. Since the y-coordinate is the same for both points, the difference in x-coordinates must be 8. Therefore, x = 10.

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22. All rhombi are parallelograms.

True

False

A rhombus is a type of quadrilateral with all sides equal in length, but it does not necessarily have parallel sides. In a parallelogram, opposite sides are parallel. Therefore, it is not true that all rhombi are parallelograms.

Explanation

A rhombus is a type of quadrilateral with all sides equal in length, but it does not necessarily have parallel sides. In a parallelogram, opposite sides are parallel. Therefore, it is not true that all rhombi are parallelograms.

23. The translation: 1 unit up and 3 units left is written as:

(x,y) --> (x + 1, y - 3)

(x,y) --> (x - 3, y + 1)

(x,y) --> (x + 3, y - 1)

(x,y) --> (x -1, y + 3)

The correct answer is (x,y) --> (x - 3, y + 1). This is because the translation is described as moving 1 unit up and 3 units left. In a Cartesian coordinate system, moving up corresponds to increasing the y-coordinate, and moving left corresponds to decreasing the x-coordinate. Therefore, the correct translation is to subtract 3 from the x-coordinate and add 1 to the y-coordinate.

Explanation

The correct answer is (x,y) --> (x - 3, y + 1). This is because the translation is described as moving 1 unit up and 3 units left. In a Cartesian coordinate system, moving up corresponds to increasing the y-coordinate, and moving left corresponds to decreasing the x-coordinate. Therefore, the correct translation is to subtract 3 from the x-coordinate and add 1 to the y-coordinate.

24. Quadrilateral HIJK has coordinates at H(-4,-12), I(6, -12), J(5, -8) and K(-5, -8). What kind of quadrilateral is it?

The given quadrilateral HIJK has opposite sides that are parallel. This can be determined by observing that the slopes of the lines HI and JK are equal, and the slopes of the lines HJ and IK are equal. Therefore, the quadrilateral satisfies the definition of a parallelogram, which is a quadrilateral with opposite sides that are parallel.

Explanation

The given quadrilateral HIJK has opposite sides that are parallel. This can be determined by observing that the slopes of the lines HI and JK are equal, and the slopes of the lines HJ and IK are equal. Therefore, the quadrilateral satisfies the definition of a parallelogram, which is a quadrilateral with opposite sides that are parallel.

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