ALGEBRA Quadrilateral ABCD Is a Rhombus. Find Each Value Or Measure

ALGEBRA Quadrilateral ABCD is a rhombus. Find each value or measure.

1. If , find . SOLUTION: A rhombus is a parallelogram with all four sides congruent. So, Then, is an isosceles triangle. Therefore, If a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles. So,

. Therefore,

2. If AB = 2x + 3 and BC = x + 7, find CD. SOLUTION: A rhombus is a parallelogram with all four sides congruent. So,

So, AB = 2(4) + 3 = 11. 6-5 Rhombi and Squares CD is congruent to AB, so CD = 11.

ALGEBRA Quadrilateral ABCD is a rhombus. COORDINATE GEOMETRY Given each set Find each value or measure. of vertices, determine whether QRST is a rhombus, a rectangle, or a square. List all that apply. Explain. 5. Q(1, 2), R(–2, –1), S(1, –4), T(4, –1) SOLUTION:

1. If , find . First graph the quadrilateral. SOLUTION: A rhombus is a parallelogram with all four sides congruent. So, Then, is an isosceles triangle. Therefore, If a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles. So,

Therefore, If the diagonals of the parallelogram are congruent, then it is a rectangle. Use the Distance Formula to 2. If AB = 2x + 3 and BC = x + 7, find CD. find the lengths of the diagonals.

SOLUTION: A rhombus is a parallelogram with all four sides congruent. So, So, the parallelogram is a rectangle. Check whether the two diagonals are perpendicular.

So, AB = 2(4) + 3 = 11. QS has a slope of . RT has a slope of . These slopes are opposite CD is congruent to AB, so CD = 11. reciprocals.

COORDINATE GEOMETRY Given each set The diagonals are perpendicular. So, it is a rhombus. of vertices, determine whether QRST is a Since the diagonals are both congruent and rhombus, a rectangle, or a square. List all that perpendicular to each other the parallelogram is a apply. Explain. rectangle, rhombus and square. 5. Q(1, 2), R(–2, –1), S(1, –4), T(4, –1) SOLUTION: 6. Q(–2, –1), R(–1, 2), S(4, 1), T(3, –2) First graph the quadrilateral. SOLUTION: First graph the quadrilateral.

If the diagonals of the parallelogram are congruent, then it is a rectangle. Use the Distance Formula to

find the lengths of the diagonals. If the diagonals of the parallelogram are congruent, eSolutions Manual - Powered by Cognero then it is a rectangle. Use the Distance Formula Pageto 1 find the lengths of the diagonals.

So, the parallelogram is a rectangle. Check whether the two diagonals are perpendicular. The diagonals are not congruent. So, the QS has a slope of . parallelogram is not a rectangle. Check whether the two diagonals are perpendicular. RT has a slope of . These slopes are opposite

reciprocals.

The diagonals are perpendicular. So, it is a rhombus. Since the diagonals are both congruent and perpendicular to each other the parallelogram is a rectangle, rhombus and square. The diagonals are not perpendicular. So, it is not a

6. Q(–2, –1), R(–1, 2), S(4, 1), T(3, –2) rhombus either. SOLUTION: ALGEBRA Quadrilateral ABCD is a rhombus. First graph the quadrilateral. Find each value or measure.

7. If AB = 14, find BC. SOLUTION: A rhombus is a parallelogram with all four sides

If the diagonals of the parallelogram are congruent, congruent. So,

then it is a rectangle. Use the Distance Formula to Therefore, BC = 14. find the lengths of the diagonals. 8. If , find . SOLUTION: The diagonals are not congruent. So, the A rhombus is a parallelogram with all four sides parallelogram is not a rectangle. Check whether the congruent. So, Then, is an two diagonals are perpendicular. isosceles triangle. Therefore, If a parallelogram is a rhombus, then each diagonal

bisects a pair of opposite angles. So,

Therefore,

The diagonals are not perpendicular. So, it is not a 9. If AP = 3x – 1 and PC = x + 9, find AC. rhombus either. SOLUTION: ALGEBRA Quadrilateral ABCD is a rhombus. The diagonals of a rhombus bisect each other. Find each value or measure. 3x – 1 = x + 9 2x = 10 x = 5 Therefore, AC = 2(5 + 9) = 28.

10. If DB = 2x – 4 and PB = 2x – 9, find PD. SOLUTION: 7. If AB = 14, find BC. The diagonals of a rhombus bisect each other. So, SOLUTION: A rhombus is a parallelogram with all four sides congruent. So, Therefore, BC = 14. Therefore, PD = PB = 2(7) – 9 = 5.

11. If , find

8. If , find . . SOLUTION: SOLUTION: A rhombus is a parallelogram with all four sides In a rhombus, consecutive interior angles are congruent. So, Then, is an supplementary. isosceles triangle. Therefore, If a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles. So,

Each pair of opposite angles of a rhombus is Therefore, congruent. So, 9. If AP = 3x – 1 and PC = x + 9, find AC. 12. If , find x. SOLUTION: The diagonals of a rhombus bisect each other. SOLUTION: 3x – 1 = x + 9 The diagonals of a rhombus are perpendicular to 2x = 10 each other. x = 5 Therefore, AC = 2(5 + 9) = 28.

10. If DB = 2x – 4 and PB = 2x – 9, find PD. SOLUTION: The diagonals of a rhombus bisect each other. So,

Therefore, PD = PB = 2(7) – 9 = 5.

11. If , find . SOLUTION: In a rhombus, consecutive interior angles are supplementary.

Each pair of opposite angles of a rhombus is congruent. So,

12. If , find x. SOLUTION: The diagonals of a rhombus are perpendicular to each other.

ALGEBRA Quadrilateral ABCD is a rhombus. Find each value or measure.

. Therefore,

2. If AB = 2x + 3 and BC = x + 7, find CD. SOLUTION: A rhombus is a parallelogram with all four sides congruent. So,

So, AB = 2(4) + 3 = 11.

CD is congruent to AB, so CD = 11.

COORDINATE GEOMETRY Given each set of vertices, determine whether QRST is a rhombus, a rectangle, or a square. List all that apply. Explain. 5. Q(1, 2), R(–2, –1), S(1, –4), T(4, –1) SOLUTION: First graph the quadrilateral.

ALGEBRA Quadrilateral ABCD is a rhombus. If the diagonals of the parallelogram are congruent, Find each value or measure. then it is a rectangle. Use the Distance Formula to find the lengths of the diagonals.

1. If , find . So, the parallelogram is a rectangle. Check whether the two diagonals are perpendicular. SOLUTION: A rhombus is a parallelogram with all four sides QS has a slope of . congruent. So, Then, is an RT has a slope of . These slopes are opposite isosceles triangle. Therefore, reciprocals. If a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles. So, The diagonals are perpendicular. So, it is a rhombus. Since the diagonals are both congruent and . perpendicular to each other the parallelogram is a rectangle, rhombus and square. Therefore,

6. Q(–2, –1), R(–1, 2), S(4, 1), T(3, –2) 2. If AB = 2x + 3 and BC = x + 7, find CD. SOLUTION: SOLUTION: First graph the quadrilateral. A rhombus is a parallelogram with all four sides congruent. So,

So, AB = 2(4) + 3 = 11.

CD is congruent to AB, so CD = 11.

COORDINATE GEOMETRY Given each set of vertices, determine whether QRST is a rhombus, a rectangle, or a square. List all that apply. Explain. If the diagonals of the parallelogram are congruent, 5. Q(1, 2), R(–2, –1), S(1, –4), T(4, –1) then it is a rectangle. Use the Distance Formula to find the lengths of the diagonals. SOLUTION: First graph the quadrilateral. The diagonals are not congruent. So, the parallelogram is not a rectangle. Check whether the two diagonals are perpendicular.

If the diagonals of the parallelogram are congruent, The diagonals are not perpendicular. So, it is not a then it is a rectangle. Use the Distance Formula to rhombus either. find the lengths of the diagonals. ALGEBRA Quadrilateral ABCD is a rhombus. Find each value or measure.

So, the parallelogram is a rectangle. Check whether the two diagonals are perpendicular. QS has a slope of . RT has a slope of . These slopes are opposite reciprocals. 7. If AB = 14, find BC.

SOLUTION: The diagonals are perpendicular. So, it is a rhombus. Since the diagonals are both congruent and A rhombus is a parallelogram with all four sides perpendicular to each other the parallelogram is a congruent. So, 6-5 Rhombirectangle, and rhombus Squares and square. Therefore, BC = 14.

6. Q(–2, –1), R(–1, 2), S(4, 1), T(3, –2) 8. If , find . SOLUTION: SOLUTION: First graph the quadrilateral. A rhombus is a parallelogram with all four sides congruent. So, Then, is an isosceles triangle. Therefore, If a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles. So,

Therefore,

9. If AP = 3x – 1 and PC = x + 9, find AC.

SOLUTION: If the diagonals of the parallelogram are congruent, then it is a rectangle. Use the Distance Formula to The diagonals of a rhombus bisect each other. find the lengths of the diagonals. 3x – 1 = x + 9 2x = 10 x = 5 Therefore, AC = 2(5 + 9) = 28. The diagonals are not congruent. So, the parallelogram is not a rectangle. Check whether the 10. If DB = 2x – 4 and PB = 2x – 9, find PD. two diagonals are perpendicular. SOLUTION: The diagonals of a rhombus bisect each other. So,

Therefore, PD = PB = 2(7) – 9 = 5. The diagonals are not perpendicular. So, it is not a rhombus either. 11. If , find . ALGEBRA Quadrilateral ABCD is a rhombus. Find each value or measure. SOLUTION: In a rhombus, consecutive interior angles are supplementary.

7. If AB = 14, find BC. Each pair of opposite angles of a rhombus is SOLUTION: congruent. So, A rhombus is a parallelogram with all four sides 12. If , find x. congruent. So, Therefore, BC = 14. SOLUTION: The diagonals of a rhombus are perpendicular to each other. 8. If , find .

SOLUTION: A rhombus is a parallelogram with all four sides eSolutionscongruent.Manual -So,Powered by Cognero Then, is an Page 2 isosceles triangle. Therefore, If a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles. So,

Therefore,

9. If AP = 3x – 1 and PC = x + 9, find AC. SOLUTION: The diagonals of a rhombus bisect each other. 3x – 1 = x + 9 2x = 10 x = 5 Therefore, AC = 2(5 + 9) = 28.

10. If DB = 2x – 4 and PB = 2x – 9, find PD. SOLUTION: The diagonals of a rhombus bisect each other. So,

Therefore, PD = PB = 2(7) – 9 = 5.

11. If , find . SOLUTION: In a rhombus, consecutive interior angles are supplementary.

Each pair of opposite angles of a rhombus is congruent. So,

12. If , find x. SOLUTION: The diagonals of a rhombus are perpendicular to each other.

ALGEBRA Quadrilateral ABCD is a rhombus. Find each value or measure.

. Therefore,

2. If AB = 2x + 3 and BC = x + 7, find CD. SOLUTION: A rhombus is a parallelogram with all four sides congruent. So,

So, AB = 2(4) + 3 = 11.

CD is congruent to AB, so CD = 11.

If the diagonals of the parallelogram are congruent, then it is a rectangle. Use the Distance Formula to find the lengths of the diagonals.

So, the parallelogram is a rectangle. Check whether the two diagonals are perpendicular. QS has a slope of . RT has a slope of . These slopes are opposite reciprocals.

The diagonals are perpendicular. So, it is a rhombus. Since the diagonals are both congruent and perpendicular to each other the parallelogram is a rectangle, rhombus and square.

6. Q(–2, –1), R(–1, 2), S(4, 1), T(3, –2) SOLUTION: First graph the quadrilateral.

If the diagonals of the parallelogram are congruent, then it is a rectangle. Use the Distance Formula to find the lengths of the diagonals.

The diagonals are not congruent. So, the parallelogram is not a rectangle. Check whether the two diagonals are perpendicular.

The diagonals are not perpendicular. So, it is not a rhombus either.

ALGEBRA Quadrilateral ABCD is a rhombus. Find each value or measure.

7. If AB = 14, find BC. SOLUTION: A rhombus is a parallelogram with all four sides congruent. So, Therefore, BC = 14.

8. If , find . SOLUTION: A rhombus is a parallelogram with all four sides congruent. So, Then, is an isosceles triangle. Therefore, If a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles. So,

Therefore,

9. If AP = 3x – 1 and PC = x + 9, find AC. SOLUTION: The diagonals of a rhombus bisect each other. 3x – 1 = x + 9 2x = 10 x = 5 Therefore, AC = 2(5 + 9) = 28.

10. If DB = 2x – 4 and PB = 2x – 9, find PD. SOLUTION: The diagonals of a rhombus bisect each other. So,

Therefore, PD = PB = 2(7) – 9 = 5.

11. If , find . SOLUTION: In a rhombus, consecutive interior angles are supplementary.

6-5 RhombiEach pair and of opposite Squares angles of a rhombus is congruent. So,

12. If , find x. SOLUTION: The diagonals of a rhombus are perpendicular to each other.