Quadrilaterals in the Coordinate Plane I can find the slope and distance between two points I can use the properties of quadrilaterals to prove that a figure in the coordinate plane is a parallelogram, rhombus, rectangle, square, or trapezoid.

Refresh… Distance: Midpoint Slope

Refresh… Parallel lines Perpendicular Lines SAME SLOPE (m) when in y=mx+b form Perpendicular Lines FLIP THE SIGN FLIP THE FRACTION

Example: Verifying Properties of Squares Show that the diagonals of square EFGH are congruent perpendicular bisectors of each other. Step 1 Show that EG and FH are congruent. Since EG = FH,

Example: Verifying Properties of Squares Show that the diagonals of square EFGH are congruent perpendicular bisectors of each other. Step 2 Show that EG and FH are perpendicular. Since ,

Example: Verifying Properties of Squares Show that the diagonals of square EFGH are congruent perpendicular bisectors of each other. Step 3 Show that EG and FH are bisect each other. Since EG and FH have the same midpoint, they bisect each other.

Example: Verifying Properties of Squares Show that the diagonals of square EFGH are congruent perpendicular bisectors of each other. The diagonals are congruent perpendicular bisectors of each other. Therefore, the quadrilateral is a SQUARE.

Your Turn: Verifying Properties of Squares The vertices of square STVW are S(–5, –4), T(0, 2), V(6, –3) , and W(1, –9) . Show that the diagonals of square STVW are congruent perpendicular bisectors of each other. Step 1 Show that SV and TW are congruent. Since SV = TW,

Your Turn: Verifying Properties of Squares The vertices of square STVW are S(–5, –4), T(0, 2), V(6, –3) , and W(1, –9) . Show that the diagonals of square STVW are congruent perpendicular bisectors of each other. Step 2 Show that SV and TW are perpendicular. Since

Your Turn: Verifying Properties of Squares The vertices of square STVW are S(–5, –4), T(0, 2), V(6, –3) , and W(1, –9) . Show that the diagonals of square STVW are congruent perpendicular bisectors of each other. Step 3 Show that SV and TW bisect each other. Since SV and TW have the same midpoint, they bisect each other. The diagonals are congruent perpendicular bisectors of each other. Therefore, the quadrilateral is a SQUARE.

Check It Out! Example 3 Continued Step 3 Show that SV and TW bisect each other. Since SV and TW have the same midpoint, they bisect each other. The diagonals are congruent perpendicular bisectors of each other.

Example Given points: A(0,2) B(3,6) C(8,6) D(5,2) How can we prove ABCD is a parallelogram, rectangle, rhombus, or a square? You MUST plot the quadrilateral for credit!!! B C A D

Example A(0,2) B(3,6) C(8,6) D(5,2) Step 1: Is ABCD a parallelogram? *check to see if slopes are parallel Therefore the quadrilateral is a Parallelogram. Slope of AB and DC are parallel Slope of Slope of BC and AD are parallel Slope of

Example A(0,2) B(3,6) C(8,6) D(5,2) Step 2: Is ABCD a rectangle or a square? *were the slopes perpendicular? AB and BC are NOT perpendicular. (slopes are not opposite reciprocals) Slope of Slope of Therefore the quadrilateral is NOT a rectangle or square. Slope of Slope of

Example A(0,2) B(3,6) C(8,6) D(5,2) Step 3: Is ABCD a rhombus? *check the lengths OR…use Pythagorean Theorem. AB and BC are equal lengths Therefore the quadrilateral is a Parallelogram AND a Rhombus.