MAFS.912.G-GPE.2.4 Use Coordinates to Prove Simple Geometric Theorems Algebraically © Copyright 2015 – all rights reserved www.cpalms.org.

MAFS.912.G-GPE.2.4 Use Coordinates to Prove Simple Geometric Theorems Algebraically
© Copyright 2015 – all rights reserved

Learning Objective: Prove or disprove that a quadrilateral in the coordinate plane defined by four given points is a: parallelogram, rectangle, rhombus, or square © Copyright 2015 – all rights reserved

What is your current level of understanding?
Level 1: I can plot points in a coordinate plane and identify polygons based on the number of sides. Level 2: I know the properties of special parallelograms and can use those properties to classify a parallelogram specifically. Level 3: I am able to use coordinates and distance formula to prove special parallelograms in the coordinate plane. Level 4: I am able to utilize the coordinate plane, properties of special parallelograms, and formulas from algebra to create problems for others to solve. © Copyright 2015 – all rights reserved

Let’s Review A quadrilateral is a four-sided polygon
A parallelogram is a quadrilateral with both pairs of opposite sides parallel Rhombi, rectangles, and squares are parallelograms with special properties © Copyright 2015 – all rights reserved

Properties of Parallelograms
Opposite sides parallel Opposite sides congruent Opposite angles congruent Consecutive angle supplementary Diagonals bisect each other Rhombi Four congruent sides Diagonals are perpendicular Diagonals bisect opposite angles Rectangles Four right angles Diagonals are congruent © Copyright 2015 – all rights reserved Squares Squares have all the properties of parallelograms, rhombi, and rectangles.

Proving That a Quadrilateral is a Parallelogram
Prove that both pairs of opposite sides are parallel Prove that both pairs of opposite sides are congruent Prove that consecutive angles are supplementary Prove that both pairs of opposite angles are congruent Prove that the diagonals bisect each other Prove that one pair of opposite sides is both congruent and parallel © Copyright 2015 – all rights reserved

Prove that both pairs of opposite sides are parallel Prove that both pairs of opposite sides are congruent Prove that consecutive angles are supplementary Prove that both pairs of opposite angles are congruent Prove that the diagonals bisect each other Prove that one pair of opposite sides is both congruent and parallel Proving That a Parallelogram is a Rhombus Prove that all four sides are congruent Prove that the diagonals are perpendicular Prove that the each diagonal bisects a pair of opposite angles © Copyright 2015 – all rights reserved

Prove that both pairs of opposite sides are parallel Prove that both pairs of opposite sides are congruent Prove that consecutive angles are supplementary Prove that both pairs of opposite angles are congruent Prove that the diagonals bisect each other Prove that one pair of opposite sides is both congruent and parallel Proving That a Parallelogram is a Rhombus Prove that all four sides are congruent Prove that the diagonals are perpendicular Prove that the each diagonal bisects a pair of opposite angles Proving That a Parallelogram is a Rectangle Prove that the diagonals are congruent Prove that all angles are right angles © Copyright 2015 – all rights reserved

Prove that both pairs of opposite sides are parallel Prove that both pairs of opposite sides are congruent Prove that consecutive angles are supplementary Prove that both pairs of opposite angles are congruent Prove that the diagonals bisect each other Prove that one pair of opposite sides is both congruent and parallel Proving That a Parallelogram is a Rhombus Prove that all four sides are congruent Prove that the diagonals are perpendicular Prove that the each diagonal bisects a pair of opposite angles Proving That a Parallelogram is a Rectangle Prove that the diagonals are congruent Prove that all angles are right angles Proving That a Parallelogram is a Square A parallelogram that is both a rhombus and a rectangle is a square © Copyright 2015 – all rights reserved

Prove that both pairs of opposite sides are parallel Prove that both pairs of opposite sides are congruent Prove that consecutive angles are supplementary Prove that both pairs of opposite angles are congruent Prove that the diagonals bisect each other Prove that one pair of opposite sides is both congruent and parallel © Copyright 2015 – all rights reserved

Prove that both pairs of opposite sides are parallel Prove that both pairs of opposite sides are congruent Prove that consecutive angles are supplementary Prove that both pairs of opposite angles are congruent Prove that the diagonals bisect each other Prove that one pair of opposite sides is both congruent and parallel Proving That a Parallelogram is a Rhombus Prove that all four sides are congruent Prove that the diagonals are perpendicular Prove that the each diagonal bisects a pair of opposite angles © Copyright 2015 – all rights reserved

Prove that both pairs of opposite sides are parallel Prove that both pairs of opposite sides are congruent Prove that consecutive angles are supplementary Prove that both pairs of opposite angles are congruent Prove that the diagonals bisect each other Prove that one pair of opposite sides is both congruent and parallel Proving That a Parallelogram is a Rhombus Prove that all four sides are congruent Prove that the diagonals are perpendicular Prove that the each diagonal bisects a pair of opposite angles Proving That a Parallelogram is a Rectangle Prove that the diagonals are congruent Prove that all angles are right angles © Copyright 2015 – all rights reserved

Prove that both pairs of opposite sides are parallel Prove that both pairs of opposite sides are congruent Prove that consecutive angles are supplementary Prove that both pairs of opposite angles are congruent Prove that the diagonals bisect each other Prove that one pair of opposite sides is both congruent and parallel Proving That a Parallelogram is a Rhombus Prove that all four sides are congruent Prove that the diagonals are perpendicular Prove that the each diagonal bisects a pair of opposite angles Proving That a Parallelogram is a Rectangle Prove that the diagonals are congruent Prove that all angles are right angles Proving That a Parallelogram is a Square A parallelogram that is both a rhombus and a rectangle is a square Prove that all four sides AND the diagonals are congruent © Copyright 2015 – all rights reserved

Determining whether a quadrilateral in the coordinate plane is a:
Parallelogram…..prove both pairs of opposite sides are congruent Rhombus……prove that all four sides are congruent Rectangle……prove that diagonals are congruent Square…….prove that all four sides are congruent and diagonals are congruent © Copyright 2015 – all rights reserved

Determining whether a quadrilateral in the coordinate plane is a:
Parallelogram…..prove both pairs of opposite sides are congruent Rhombus……prove that all four sides are congruent Rectangle……prove that diagonals are congruent Square…….prove that all four sides are congruent and diagonals are congruent Example 1: Determine whether the quadrilateral is a parallelogram, rhombus, rectangle, square, or not a parallelogram. B (3,2) A (4,0) R (0,0) T (-1,2) © Copyright 2015 – all rights reserved

square, or not a parallelogram. B (3,2) A (4,0) R (0,0) T (-1,2)
Example 1: Determine whether the quadrilateral is a parallelogram, rhombus, rectangle, square, or not a parallelogram. B (3,2) A (4,0) R (0,0) T (-1,2) © Copyright 2015 – all rights reserved

Example 1: Determine whether the quadrilateral is a parallelogram, rhombus, rectangle, square, or not a parallelogram. B (3,2) A (4,0) R (0,0) T (-1,2) Sides: BA = (3−4) 2 + (2−0) 2 = 1+4 = 5 AR = (4−0) 2 + (0−0) 2 = = 16 =4 RT = (0−(−1)) 2 + (0−2) 2 = 1+4 = 5 TB = (−1−3) 2 + (2−2) 2 = = 16 =4 © Copyright 2015 – all rights reserved

Example 1: Determine whether the quadrilateral is a parallelogram, rhombus, rectangle, square, or not a parallelogram. B (3,2) A (4,0) R (0,0) T (-1,2) Sides: BA = (3−4) 2 + (2−0) 2 = 1+4 = 5 AR = (4−0) 2 + (0−0) 2 = = 16 =4 RT = (0−(−1)) 2 + (0−2) 2 = 1+4 = 5 TB = (−1−3) 2 + (2−2) 2 = = 16 =4 Diagonals: BR = (3−0) 2 + (2−0) 2 = 9+4 = 13 AT = (4−(−1)) 2 + (−1−2) 2 = = 34 © Copyright 2015 – all rights reserved

Example 1: Determine whether the quadrilateral is a parallelogram, rhombus, rectangle, square, or not a parallelogram. B (3,2) A (4,0) R (0,0) T (-1,2) Sides: BA = (3−4) 2 + (2−0) 2 = 1+4 = 5 AR = (4−0) 2 + (0−0) 2 = = 16 =4 RT = (0−(−1)) 2 + (0−2) 2 = 1+4 = 5 TB = (−1−3) 2 + (2−2) 2 = = 16 =4 Diagonals: BR = (3−0) 2 + (2−0) 2 = 9+4 = 13 AT = (4−(−1)) 2 + (−1−2) 2 = = 34 Since opposite sides are congruent, but diagonals are not congruent, the quadrilateral is a parallelogram. © Copyright 2015 – all rights reserved

square, or not a parallelogram. G (4,0) I (3,2) R (-3,-1) L (-2,-3)
Example 2: Determine whether the quadrilateral is a parallelogram, rhombus, rectangle, square, or not a parallelogram. G (4,0) I (3,2) R (-3,-1) L (-2,-3) © Copyright 2015 – all rights reserved

Example 2: Determine whether the quadrilateral is a parallelogram, rhombus, rectangle, square, or not a parallelogram. G (4,0) I (3,2) R (-3,-1) L (-2,-3) Sides: GI = (4−3) 2 + (0−2) 2 = 1+4 = 5 IR = (3− −3 ) 2 + (2−(−1)) 2 = = 45 =3 5 RL = (−3−(−2)) 2 + (−1−(−3)) 2 = 1+4 = 5 LG = (−2−4) 2 + (−3−0) 2 = = 45 =3 5 © Copyright 2015 – all rights reserved

Example 2: Determine whether the quadrilateral is a parallelogram, rhombus, rectangle, square, or not a parallelogram. G (4,0) I (3,2) R (-3,-1) L (-2,-3) Sides: GI = (4−3) 2 + (0−2) 2 = 1+4 = 5 IR = (3− −3 ) 2 + (2−(−1)) 2 = = 45 =3 5 RL = (−3−(−2)) 2 + (−1−(−3)) 2 = 1+4 = 5 LG = (−2−4) 2 + (−3−0) 2 = = 45 =3 5 Diagonals: GR = (4− −3 ) 2 + (0−(−1)) 2 = = 50 =5 2 IL = (3−(−2)) 2 + (2−(−3)) 2 = = 50 =5 2 © Copyright 2015 – all rights reserved

Example 2: Determine whether the quadrilateral is a parallelogram, rhombus, rectangle, square, or not a parallelogram. G (4,0) I (3,2) R (-3,-1) L (-2,-3) Sides: GI = (4−3) 2 + (0−2) 2 = 1+4 = 5 IR = (3− −3 ) 2 + (2−(−1)) 2 = = 45 =3 5 RL = (−3−(−2)) 2 + (−1−(−3)) 2 = 1+4 = 5 LG = (−2−4) 2 + (−3−0) 2 = = 45 =3 5 Diagonals: GR = (4− −3 ) 2 + (0−(−1)) 2 = = 50 =5 2 IL = (3−(−2)) 2 + (2−(−3)) 2 = = 50 =5 2 Since opposite sides are congruent, and diagonals are congruent, the quadrilateral is a rectangle. © Copyright 2015 – all rights reserved

square, or not a parallelogram. W (3,1) X (5,2) Y (3,3) Z (1,2)
Example 3: Determine whether the quadrilateral is a parallelogram, rhombus, rectangle, square, or not a parallelogram. W (3,1) X (5,2) Y (3,3) Z (1,2) © Copyright 2015 – all rights reserved

square, or not a parallelogram. W (3,1) X (5,2) Y (3,3) Z (1,2) Sides:
Example 3: Determine whether the quadrilateral is a parallelogram, rhombus, rectangle, square, or not a parallelogram. W (3,1) X (5,2) Y (3,3) Z (1,2) Sides: WX = (3−5) 2 + (1−2) 2 = 4+1 = 5 XY = (5−3) 2 + (2−3) 2 = 4+1 = 5 YZ = (3−1) 2 + (3−2) 2 = 4+1 = 5 ZW = (1−3) 2 + (2−1) 2 = 4+1 = 5 © Copyright 2015 – all rights reserved

Example 3: Determine whether the quadrilateral is a parallelogram, rhombus, rectangle, square, or not a parallelogram. W (3,1) X (5,2) Y (3,3) Z (1,2) Sides: WX = (3−5) 2 + (1−2) 2 = 4+1 = 5 XY = (5−3) 2 + (2−3) 2 = 4+1 = 5 YZ = (3−1) 2 + (3−2) 2 = 4+1 = 5 ZW = (1−3) 2 + (2−1) 2 = 4+1 = 5 Diagonals: WY = (3−3) 2 + (1−3) 2 = 0+4 =2 XZ = (5−1) 2 + (2−2) 2 = =4 © Copyright 2015 – all rights reserved

Example 3: Determine whether the quadrilateral is a parallelogram, rhombus, rectangle, square, or not a parallelogram. W (3,1) X (5,2) Y (3,3) Z (1,2) Sides: WX = (3−5) 2 + (1−2) 2 = 4+1 = 5 XY = (5−3) 2 + (2−3) 2 = 4+1 = 5 YZ = (3−1) 2 + (3−2) 2 = 4+1 = 5 ZW = (1−3) 2 + (2−1) 2 = 4+1 = 5 Diagonals: WY = (3−3) 2 + (1−3) 2 = 0+4 =2 XZ = (5−1) 2 + (2−2) 2 = =4 Since all four sides are congruent, but diagonals are not congruent, the quadrilateral is a rhombus. © Copyright 2015 – all rights reserved

square, or not a parallelogram. J (-3,0) K (0,3) L (3,0) M (0,-3)
Example 4: Determine whether the quadrilateral is a parallelogram, rhombus, rectangle, square, or not a parallelogram. J (-3,0) K (0,3) L (3,0) M (0,-3) © Copyright 2015 – all rights reserved

Example 4: Determine whether the quadrilateral is a parallelogram, rhombus, rectangle, square, or not a parallelogram. J (-3,0) K (0,3) L (3,0) M (0,-3) Sides: JK = (−3−0) 2 + (0−3) 2 = 9+9 = 18 =3 2 KL = (0−3) 2 + (3−0) 2 = 9+9 = 18 =3 2 LM = (3−0) 2 + (0−(−3)) 2 = 9+9 = 18 =3 2 MJ = (0−(−3)) 2 + (−3−0) 2 = 9+9 = 18 =3 2 © Copyright 2015 – all rights reserved

Example 4: Determine whether the quadrilateral is a parallelogram, rhombus, rectangle, square, or not a parallelogram. J (-3,0) K (0,3) L (3,0) M (0,-3) Sides: JK = (−3−0) 2 + (0−3) 2 = 9+9 = 18 =3 2 KL = (0−3) 2 + (3−0) 2 = 9+9 = 18 =3 2 LM = (3−0) 2 + (0−(−3)) 2 = 9+9 = 18 =3 2 MJ = (0−(−3)) 2 + (−3−0) 2 = 9+9 = 18 =3 2 Diagonals: JL = (−3−3) 2 + (0−0) 2 = 36 =6 KM = (0−0) 2 + (3−(−3)) 2 = 36 =6 © Copyright 2015 – all rights reserved

Example 4: Determine whether the quadrilateral is a parallelogram, rhombus, rectangle, square, or not a parallelogram. J (-3,0) K (0,3) L (3,0) M (0,-3) Sides: JK = (−3−0) 2 + (0−3) 2 = 9+9 = 18 =3 2 KL = (0−3) 2 + (3−0) 2 = 9+9 = 18 =3 2 LM = (3−0) 2 + (0−(−3)) 2 = 9+9 = 18 =3 2 MJ = (0−(−3)) 2 + (−3−0) 2 = 9+9 = 18 =3 2 Diagonals: JL = (−3−3) 2 + (0−0) 2 = 36 =6 KM = (0−0) 2 + (3−(−3)) 2 = 36 =6 Since all four sides are congruent, and diagonals are congruent, the quadrilateral is a square. © Copyright 2015 – all rights reserved

square, or not a parallelogram. Q (0,0) R (6,0) S (9,1) T (3,2)
Example 5: Determine whether the quadrilateral is a parallelogram, rhombus, rectangle, square, or not a parallelogram. Q (0,0) R (6,0) S (9,1) T (3,2) © Copyright 2015 – all rights reserved

square, or not a parallelogram. Q (0,0) R (6,0) S (9,1) T (3,2) Sides:
Example 5: Determine whether the quadrilateral is a parallelogram, rhombus, rectangle, square, or not a parallelogram. Q (0,0) R (6,0) S (9,1) T (3,2) Sides: QR = (0−6) 2 + (0−0) 2 = =6 RS= (6−9) 2 + (0−1) 2 = 9+1 = 10 ST = (9−3) 2 + (1−2) 2 = = 37 TQ = (3−0) 2 + (2−0) 2 = 9+4 = 13 © Copyright 2015 – all rights reserved

square, or not a parallelogram. Q (0,0) R (6,0) S (9,1) T (3,2) Sides:
Example 5: Determine whether the quadrilateral is a parallelogram, rhombus, rectangle, square, or not a parallelogram. Q (0,0) R (6,0) S (9,1) T (3,2) Sides: QR = (0−6) 2 + (0−0) 2 = =6 RS= (6−9) 2 + (0−1) 2 = 9+1 = 10 ST = (9−3) 2 + (1−2) 2 = = 37 TQ = (3−0) 2 + (2−0) 2 = 9+4 = 13 Since opposite sides ARE NOT congruent the quadrilateral is a not a parallelogram. © Copyright 2015 – all rights reserved

Your turn! Use the distance formula and the given coordinates to compute the lengths of the sides and diagonals of the quadrilateral. Then, compare the lengths to determine whether the quadrilateral is a: Parallelogram Rhombus Rectangle Square Not a parallelogram © Copyright 2015 – all rights reserved

Determine whether the quadrilateral is a parallelogram, rhombus, rectangle,
square, or not a parallelogram. Q (3,-4) R (1,1) S (4,2) T (6,-3) Sides: QR = ? RS = ? ST = ? TQ = ? Diagonals: QS = ? RT = ? © Copyright 2015 – all rights reserved

Q (3,-4) R (1,1) S (4,2) T (6,-3) Sides: QR = (3−1) 2 + (−4−1) 2 = 29
TQ = (6−3) 2 + (−3+4) 2 = 10 Diagonals: QS = (3−4) 2 + (−4−2) 2 = 37 RT = (1−6) 2 + (1+3) 2 = 41 © Copyright 2015 – all rights reserved

Q (3,-4) R (1,1) S (4,2) T (6,-3) Sides: QR = (3−1) 2 + (−4−1) 2 = 29
TQ = (6−3) 2 + (−3+4) 2 = 10 Diagonals: QS = (3−4) 2 + (−4−2) 2 = 37 RT = (1−6) 2 + (1+3) 2 = 41 © Copyright 2015 – all rights reserved Since opposite sides ARE congruent, but diagonals ARE NOT congruent, the quadrilateral is a PARALLELOGRAM.

U (1,2) V (-3,-1) W (-3,4) X (1,7) Sides: UV = ? VW = ? WX = ? XU = ? Diagonals: UW = ? VX = ? Determine whether the quadrilateral is a parallelogram, rhombus, rectangle, square, or not a parallelogram. © Copyright 2015 – all rights reserved

U (1,2) V (-3,-1) W (-3,4) X (1,7) Sides: UV = (1+3) 2 + (2+1) 2 =5
XU = (1−1) 2 + (7−2) 2 =5 Diagonals: UW = (1+3) 2 + (2+1) 2 =2 5 VX = (−3−1) 2 + (−1−7) 2 =4 5 © Copyright 2015 – all rights reserved

U (1,2) V (-3,-1) W (-3,4) X (1,7) Sides: UV = (1+3) 2 + (2+1) 2 =5
XU = (1−1) 2 + (7−2) 2 =5 Diagonals: UW = (1+3) 2 + (2+1) 2 =2 5 VX = (−3−1) 2 + (−1−7) 2 =4 5 © Copyright 2015 – all rights reserved Since all four sides ARE congruent, but diagonals ARE NOT congruent, the quadrilateral is a RHOMBUS.

Wrapping Things Up You can use algebraic formulas to prove geometric theorems. Today, we used the distance formula to find the lengths of the sides and diagonals of quadrilaterals in the coordinate plane. We compared the lengths of the sides and diagonals to prove that the quadrilateral in the coordinate plane was a parallelogram, a rhombus, a rectangle, a square, or not a parallelogram. © Copyright 2015 – all rights reserved

SUMMARY Proving That a Quadrilateral is a Parallelogram
Prove that both pairs of opposite sides are parallel Prove that both pairs of opposite sides are congruent Prove that consecutive angles are supplementary Prove that both pairs of opposite angles are congruent Prove that the diagonals bisect each other Prove that one pair of opposite sides is both congruent and parallel Proving That a Parallelogram is a Rhombus Prove that all four sides are congruent Prove that the diagonals are perpendicular Prove that the each diagonal bisects a pair of opposite angles Proving That a Parallelogram is a Rectangle Prove that the diagonals are congruent Prove that all angles are right angles Proving That a Parallelogram is a Square A parallelogram that is both a rhombus and a rectangle is a square Prove that all four sides AND the diagonals are congruent © Copyright 2015 – all rights reserved

MAFS.912.G-GPE.2.4 Use Coordinates to Prove Simple Geometric Theorems Algebraically © Copyright 2015 – all rights reserved www.cpalms.org.

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MAFS.912.G-GPE.2.4 Use Coordinates to Prove Simple Geometric Theorems Algebraically © Copyright 2015 – all rights reserved www.cpalms.org.

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