Parallelogram, rectangle, rhombus and squares G-28 “I can use coordinates to prove geometric theorems algebraically.” Parallelogram, rectangle, rhombus and squares

Remember Regular parallelogram: Opposite sides are parallel Opposite sides are congruent One pair of sides are parallel and congruent. When you are working with coordinates, always graph the points first.

Ex1: Three vertices of parallelogram JKLM are J(3, –8), K(–2, 2), and L(2, 6). Find the coordinates of vertex M.

Three vertices of PQSR are P(–3, –2), Q(–1, 4), and S(5, 0) Three vertices of PQSR are P(–3, –2), Q(–1, 4), and S(5, 0). Find the coordinates of vertex R.

Ex 2: JKLM is a parallelogram. J(–1, –6), K(–4, –1), L(4, 5), M(7, 0) Ex 2: JKLM is a parallelogram. J(–1, –6), K(–4, –1), L(4, 5), M(7, 0). Prove that opposite sides are parallel. (parallel lines have SAME Slope)

Ex 3: ABCD is a parallelogram. A(2, 3), B(6, 2), C(5,0), D(1, 1) Ex 3: ABCD is a parallelogram. A(2, 3), B(6, 2), C(5,0), D(1, 1). Prove that 1 pair of opposite sides are parallel and congruent.

Ex 3 cont: ABCD is a parallelogram. A(2, 3), B(6, 2), C(5, 0), D(1, 1) Ex 3 cont: ABCD is a parallelogram. A(2, 3), B(6, 2), C(5, 0), D(1, 1). Prove that 1 pair of opposite sides are parallel and congruent.

A square is a quadrilateral with four right angles and four congruent sides. In the exercises, you will show that a square is a parallelogram, a rectangle, and a rhombus. So a square has the properties of all three.

To show that the diagonals of a square are perpendicular bisectors. Diagonals are perpendicular (find slopes, and they need to be opposite reciprocal) Diagonals are congruent (distance formula and they need to be the same) Diagonals have same midpoint

Ex 4: Show that the diagonals of square EFGH are congruent perpendicular bisectors of each other. E(-4, -1), F(-1, 3), G(3, 0), H(0, -4)

Ex 4 cont: Show that the diagonals of square EFGH are congruent perpendicular bisectors of each other. E(-4, -1), F(-1, 3), G(3, 0), H(0, -4)

Ex 5: Show that the diagonals of square STVW are congruent perpendicular bisectors of each other. S(–5, –4), T(0, 2), V(6, –3) ,W(1, –9)

Ex 5 cont: Show that the diagonals of square STVW are congruent perpendicular bisectors of each other. S(–5, –4), T(0, 2), V(6, –3) ,W(1, –9)

Ex 6: Use the diagonals to determine whether a parallelogram w/the given vertices is a rectangle, rhombus, or square. Give all the names that apply. A(0, 2) B(3, 6) C(8, 6) D(5, 2) [1] Graph Note: if a quadrilateral is both a rectangle and rhombus, then it is a square.

A(0, 2) B(3, 6) C(8, 6) D(5, 2) [2] Determine if ABCD is a rectangle (diagonals are congruent)

A(0, 2) B(3, 6) C(8, 6) D(5, 2) [3] Determine if ABCD is a rhombus (diagonals are perpendicular; slope is opposite reciprocal)

Ex 7: Use the diagonals to determine whether a parallelogram with the given vertices is a rectangle, rhombus, or square. Give all the names that apply. P(–1, 4), Q(2, 6), R(4, 3), S(1, 1)

P(–1, 4), Q(2, 6), R(4, 3), S(1, 1) [2] Determine if PQRS is a rectangle (diagonals are congruent)

P(–1, 4), Q(2, 6), R(4, 3), S(1, 1) [3] Determine if PQRS is a rhombus (diagonals are perpendicular; slope is opposite reciprocal)