Proving Properties of a Parallelogram

Transcription

1 Eplain Proving Properties of a Parallelogram You have alread used the Distance Formula and the Midpoint Formula in coordinate proofs As ou will see, slope is useful in coordinate proofs whenever ou need to show that lines are parallel or perpendicular Eample Prove or disprove that the quadrilateral determined b the points A (4, 4), B (3, ), C (, ), and D (, ) is a parallelogram Use slopes to write the coordinate proof To determine whether ABCD is a parallelogram, find the slope of each side of the quadrilateral Slope of AB ; Slope of BC ; Slope of CD (-) - - (-) 3 3; Slope of DA (-) 5 Compare slopes The slopes of opposite sides are equal This means opposite sides are parallel So, quadrilateral ABCD is a parallelogram -4 C D 4 A B 0 4 Module Lesson 4

2 B Use the Distance Formula to write the coordinate proof To determine whether ABCD is a parallelogram, find the length of each side of the quadrilateral Remember that the Distance Formula is length ( - ) + ( - ) AB ( - 4) + ( - 4) BC 3 (- - 3 ) - (-) + ( ) -3 (-5) + ( - ) ( - ) CD (- - - ) () + ( 3 ) 0 + ( - (-)) DA (4 - - ) + (4 - ) ( 5 ) + ( ) 9 Compare the side lengths The lengths of the opposite sides are equal B the Opposites Sides Criterion for a Parallelogram, we can conclude that ABCD is a parallelogram Reflect 3 Suppose ou want to prove that a general parallelogram WXYZ has diagonals that bisect each other Wh is it convenient to use general verte coefficients, such as a and b? To prove that the diagonals bisect each other, ou can show that the diagonals share the same midpoint Because the midpoint formula involves dividing b, if the vertices have a coefficient of, then the result will be a whole number Your Turn Write a coordinate proof given quadrilateral ABCD with vertices A (3, ), B (8, ), C (5, 0), and D (0, 0) 4 Prove that ABCD is a parallelogram Possible answer: AB (8-3) + ( - ) 5; DC AD (3-0) + ( - 0) 3 ; BC 5 Prove that the diagonals of ABCD bisect each other (5-0) + (0-0) 5 (8-5) + ( - 0) 3 Since the opposite sides have the same lengths, ABCD is a parallelogram Show that the diagonals share the same midpoint Midpoint of DB : ( 0 + 8, 0 + ) (4, ) Midpoint of AC : (, + 0 ) (4, ) Since the diagonals share a midpoint, the bisect each other Module Lesson 4

3 Eplain Proving Conditions for Special Parallelograms Eample Prove or disprove each statement about the quadrilateral determined b the points Q (, -3), R (-4, 0), S (-, 4), and T (4, ) The diagonals of QRST are congruent The length of SQ The length of RT 65 (-4-4) + (0 - ) 65 ( - (-) ) + (-3-4) So, the diagonals of QRST are congruent QRST is a rectangle Find the slope of each side of the quadrilateral Slope of QR (-3) ; Slope of RS ; Slope of ST ; R -4 S 4 T Q Slope of TQ Find the products of the slopes of adjacent sides (slope of QR ) (slope of RS ) - - ; (slope of RS ) (slope of ST ) - ; - (slope of ST ) (slope of TQ ) - ; (slope of TQ ) (slope of QR - ) - - You can conclude that adjacent sides are perpendicular So, quadrilateral QRST is a rectangle Reflect 6 Eplain how to prove that QRST is not a square Use the Distance Formula to compare adjacent sides; because the sides are not all congruent, QRST is not a square Module Lesson 4

4 Your Turn Prove or disprove each statement about quadrilateral WXYZ determined b the points W (0, 0), X (4, 3), Y (9, 3), and Z (5, 0) 4 W Z X Y 7 WXYZ is a rhombus 8 The diagonals of WXYZ are perpendicular WX (4-0) + (3-0) 5 5 XY (9-4) + (3-3) Slope of WY YZ (5-9) + (0-3) Slope of XZ ZW (0-5) + (0-0) (Slope of WY )(Slope of XZ ) So, WY is perpendicular to XZ Since all four sides have the same length, WXYZ is a rhombus Eplain 3 Identifing Figures on the Coordinate Plane Eample 3 Use the diagonals to determine whether a parallelogram with the given vertices is a rectangle, rhombus, or square Give all the names that appl A (0, ), B (3, 6), C (8, 6), D (5, ) Step Graph ABCD Step Determine if ABCD is a rectangle AC (8-0) + (6 - ) BD (5-3) + ( - 6) 0 5 Since 4 5 5, ABCD is not a rectangle Thus, ABCD is not a square Step 3 Determine if ABCD is a rhombus Slope of AC Slope of BD Since ( ) (-) -, AC BD ABCD is a rhombus 6 B C 4 A D Module 0 54 Lesson 4

5 B E( 4, ), F( 3, ), G(3, 0), H(, 3) Step Graph EFGH Step Determine if EFGH is a rectangle EG ( 3 - ) + ( 0 - ) FH ( - (-3)) + ( - ) 5 E Since 5 5, the diagonals are congruent EFGH is a rectangle F G H Step 3 Determine if EFGH is a rhombus Slope of 0 - (-) 3 - (-4) EG 7 ; Slope of FH (- 3) Since ( 7) (-) -, EG is not perpendicular to FH So, EFGH is not a rhombus and cannot be a square Your Turn Use the diagonals to determine whether a parallelogram with the given vertices is a rectangle, rhombus, or square Give all the names that appl 9 K( 5, ), L(, 4), M(3, ), N(0, 4) 0 P (-4, 6), Q (, 5), R (3, -), S (-3, 0) (0 - (-)) + (-4-4) 7 (3 - (-5)) + ( - (-)) 7 LN KM Since 7 7, the diagonals are congruent; KLMN is a rectangle Slope of -4-4 LN 0 - (-) -4 Slope of - (-) KM 3 - (-5) 4 Since ( -4 ) ( 4 ) -, the diagonals are perpendicular; KLMN is a rhombus Since KLMN is both a rectangle and a rhombus, it is also a square Elaborate (3 - (-4)) + (- - 6) 7 PR QS (-3 - ) + (0-5) 5 Since 7 5, the diagonals are not congruent So PQRS is not a rectangle, and thus not a square Slope of PR (-4) - Slope of QS Since (-)() -, the diagonals are perpendicular, so PQRS is a rhombus How can ou use slopes to show that two line segments are parallel? Perpendicular? Parallel lines have the same slopes Perpendicular lines have slopes that are opposite inverses, so when ou multipl them, the product will be - When ou use the distance formula, ou find the square root of a value When finding the square root of a value, ou must consider both the positive and negative outcomes Eplain wh the negative outcome is not used in the coordinate proofs in the lesson Length is alwas a positive number, so the negative result of evaluating the square root does not appl to the situation Module 0 54 Lesson 4