Day 107 – A rectangle and a square on the x-y plane

Introduction A plane figure in the x-y plane can resemble one of the common quadrilaterals we have come across, but that is not enough to show that the figure meets all the basic properties that define the quadrilateral we think it is. We need to further prove algebraically that the figure has all the properties of the quadrilateral we think it is. In this lesson, we will determine the properties that can be used to identify a square and a rectangle on the x-y plane. We will further learn how to determine the coordinates of the missing vertex that makes up a rectangle and a square.

Vocabulary 1. x-y plane The plane formed by the intersection of the horizontal axis, referred to as the x-axis and the vertical axis referred to as the y-axis. 2. A rectangle A quadrilateral whose four interior angles are all right angles and whose opposite sides are parallel and congruent. 3. A square A quadrilateral whose four interior angles are right angles and whose sides are all congruent.

4. Coordinate geometry A special kind of geometry where the position of points on the plane is given in terms of an ordered pair of numbers and coordinates are used to find measurements on plane figures on the 𝑥−𝑦 plane. It is also referred to as analytic geometry.

We have already learned how to find the slope of a line segment on the x-y plane. In coordinate geometry, the slope is used to show that lines are either parallel or perpendicular. We should recall that: 1. Parallel lines have the same slope 2. Perpendicular lines have slopes such that the slopes are negative reciprocals of each other. We will also use the distance formula to find the distance between points and also to show that two sides of a quadrilateral are congruent.

The distance formula The distance, 𝑑 between two points on the x-y plane with coordinates 𝑥 1 , 𝑦 1 and 𝑥 2 , 𝑦 2 is given by: 𝑑= 𝑥 2 − 𝑥 1 2 + 𝑦 2 − 𝑦 1 2

Basic properties of a square 1. All the four sides are equal 2. The diagonals are equal 3. All the four angles are equal, each measuring 90° 4. Opposite sides are congruent. 5. Opposite sides are parallel.

Basic properties of a rectangle 1. It has two pairs of equal sides 2. Opposite sides are parallel 3. Opposite sides are equal 4. 3.All the four angles are equal, each measuring 90° 5. The diagonals are equal

Showing that a quadrilateral is a square on the x-y plane. To determine the properties that can be used to show that a quadrilateral is a square on the x-y plane, we use the following procedure: 1. We show that the quadrilateral is a rhombus by proving that all the four sides are congruent. 2. We then show that the quadrilateral is a rectangle by proving that its diagonals are congruent.

Example 1 Show that a quadrilateral with coordinates A 1,1 , B 4,1 , C 4,4 and D 1,4 is a square. Solution 1.We show that the quadrilateral is a rhombus by proving that all the four sides are congruent. We use the distance formula to calculate the lengths of the sides of the quadrilateral. 𝐴𝐵= 4−1 2 + 1−1 2 = 9 =3 units 𝐵𝐶= 4−4 2 + 4−1 2 = 9 =3 units 𝐶𝐷= 4−1 2 + 4−4 2 = 9 =3 units 𝐴𝐷= 1−1 2 + 4−4 2 = 9 =3 units

The quadrilateral is a rhombus since all the sides are congruent. 2 The quadrilateral is a rhombus since all the sides are congruent. 2. We then show that the quadrilateral is a rectangle by proving that its diagonals are congruent. The diagonals will be AC and BD. We use the distance formula to show that the diagonals are congruent. 𝐴𝐶= 4−1 2 + 4−1 2 = 18 units 𝐵𝐷= 4−1 2 + 1−4 2 = 18 units The diagonals are congruent hence the quadrilateral is a square.

Showing that a quadrilateral is a rectangle on the x-y plane. To determine the properties that can be used to show that a quadrilateral is a rectangle on the x-y plane, we use the following procedure: 1.We show that the quadrilateral is a parallelogram by proving that both pairs of opposite sides are congruent by calculating the distances of all four sides. 2. We then show that the quadrilateral is a rectangle by proving that its diagonals are congruent.

Example 2 Show that a quadrilateral with coordinates; K 4,3 , L 11,3 , M 4,7 and N 11,7 is a rectangle. Solution 1.We show that the quadrilateral is a parallelogram by proving that both pairs of opposite sides are congruent by calculating the distances of all four sides. In parallelogram KLMN the pairs of opposite sides are KL and MN; KN and LM. We use the distance formula to find the length of these opposite sides.

We show that the first pair of opposite sides, KL and MN contains equal sides. 𝐾𝐿= 11−4 2 + 3−3 2 = 49 =7 units 𝑀𝑁= 11−4 2 + 7−7 2 = 49 =7 units These opposite sides are congruent. We then show that the second pair of opposite sides, KN and LM contains equal angles. 𝐾𝑁= 11−4 2 + 7−3 2 = 65 units 𝐿𝑀= 11−4 2 + 3−7 2 = 65 units These opposite sides are also congruent. This shows that the quadrilateral is a parallelogram.

2. We then show that the quadrilateral is a rectangle by proving that its diagonals are congruent. The diagonals are KM and LN. 𝐾𝑀= 4−4 2 + 7−3 2 = 16 =4 units 𝐿𝑁= 11−11 2 + 7−3 2 = 16 =4 units The diagonals are congruent hence quadrilateral KLMN is a rectangle.

Determining the coordinates of the missing vertex that makes up a square and a rectangle In coordinate geometry, we can also use the properties of a given quadrilateral to find the coordinates of a missing vertex when we are given the coordinates of the other three vertices. We will learn how to do find the coordinates of the missing vertex that make up a rectangle and a square by use of an example.

Example 3 A rectangle WXYZ has three of its coordinates given as: W 1,1 , Y 5,3 and 𝑍 1,3 . Find the coordinates of Y. Solution We can let the coordinates of Y to be the coordinates of an arbitrary point 𝑌 𝑥,𝑦 . The sides of the rectangle are WX parallel to YZ and WZ parallel to XY. A rectangle has parallel opposite sides and this means that the opposite sides have the same slope.

This implies that the slope of WX should be equal to the slope of YZ This implies that the slope of WX should be equal to the slope of YZ. slope of WX=slope of YZ 1−1 5−1 = 𝑦−3 𝑥−1 After cross-multiplying, we have: 4 𝑦−3 =0 4𝑦−12=0 𝑦=3 Similarly, the slope of WZ should be equal to the slope of XY slope of WZ=slope of XY 3−1 1−1 = 𝑦−1 𝑥−5

After cross-multiplying, we have: 2 𝑥−5 =0 2𝑥−10=0 𝑥=5 We have found the values of x and y, the coordinates of vertex Y will be given as 𝒀 𝟓,𝟑

homework A quadrilateral JKLM has coordinates; J −1,1 , K 4,1 , L 4,3 and M −1,3 . (a) Find the distance of each side in the following pairs of opposite sides: (i) JK and LM (ii) JM and KL (b) What do you notice about the length of the sides in each pair of opposite sides in (a) above? (c) Basing on your response in (b) above, which type of quadrilateral is JKLM likely to be?

(d) Identify the diagonals of the quadrilateral JKLM (d) Identify the diagonals of the quadrilateral JKLM. (e) Calculate the length of the diagonals you have identified in (d) above. (f) What do you notice about the lengths of the diagonals in (e). (g) Basing on your response in (f) above, what type of quadrilateral should JKLM be?

Answers to homework (a) (i) JK=5 units LM=5 units (ii) JM=2 units KL=2 units (b) The opposite sides have the same length in each pair (c) A parallelogram (d) KM and JL (e) KM= 29 units JL= 29 units

(f) The diagonals have the same length. (g) A rectangle

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