Day 109 – A trapezoid on the x-y plane

Introduction A plane figure plotted on the x-y plane can be seen to have the appearance of a trapezoid, but that is barely enough to show that the figure has all the major properties of a trapezoid. The figure might be a trapezoid or any other quadrilateral. We need to further prove algebraically that the given figure has all the basic properties that define a trapezoid in order to verify this. In this lesson, we will determine the properties that can be used to identify a trapezoid on the x-y plane. We will further learn how to determine the coordinates of the missing vertex that makes up a trapezoid.

Vocabulary 1. 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 2. Trapezoid A quadrilateral that has at least a pair of parallel opposite sides. 3. Isosceles trapezoid A special trapezoid whose pair of non-parallel sides are congruent.

A trapezoid Quadrilateral FGHI below is a trapezoid. FI and GH are the parallel sides called the bases whereas FG and HI are the non-parallel sides called legs. FG∥HI F G H I

An isosceles trapezoid Quadrilateral ABCD below is an isosceles trapezoid. The diagonals AC and BD are congruent. ∠𝐴𝐵𝐶≅∠𝐵𝐶𝐷 AD and BC are the bases where as AB and CD are the legs. 𝐴𝐷∥𝐵𝐶 and 𝐴𝐵 ≅ 𝐶𝐷 A B C D

Basic properties of an Isosceles trapezoid 1. It has one pair of parallel sides, called the bases 2. It has one pair of equal sides, called the legs 3. Its two base angles are equal 4. Its diagonals are congruent 5. Its opposite interior angles add up to 180°, i.e they are supplementary

In order to show that a quadrilateral is a trapezoid In order to show that a quadrilateral is a trapezoid. We need to recall the following keys concepts that will be used in the course of our proof: 1. 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

2. Slope of a straight line formula: The slope, m of a straight line passing through two points on the x-y plane with coordinates 𝑥 1 , 𝑦 1 and 𝑥 2 , 𝑦 2 is given by: 𝑚= 𝑦 2 − 𝑦 1 𝑥 2 − 𝑥 1 We should also recall that parallel lines have the same slope.

Showing that a quadrilateral is a trapezoid on the x-y plane. In order to show that a given quadrilateral is a trapezoid: 1. We show that one pair of opposite sides are parallel. They will have the same slope. 2. We then show that the other pair of opposite sides is not parallel. They will have different slopes. We have to calculate the slope of all the sides in order then identify the pair of parallel sides and the pair of non-parallel sides.

Example 1 Show that a quadrilateral FGHI whose vertices have the coordinates given as 𝐹 −2, 3 , 𝐺 2, −1 , 𝐻 4, 3 and 𝐼 2, 5 is a trapezoid. Solution We first calculate the slope of the fours sides of the quadrilateral. Slope of FG= 3+1 −2−2 = 4 −4 =−1

Slope of GH= −1−3 2−4 = −4 −2 =2 Slope of HI= 3−5 4−2 = −2 2 =−1 Slope of FI= 3−5 −2−2 = −2 −4 = 1 2 The slopes show that 𝐹𝐺∥𝐻𝐼 and 𝐹𝐼∦𝐺𝐻 Quadrilateral FGHI is a trapezoid because one pair of opposite sides is parallel and the other pair of opposite sides is not parallel.

Showing that a quadrilateral is an isosceles trapezoid on the x-y plane. In order to show that a given quadrilateral is an isosceles trapezoid: 1. We first show that it is a trapezoid by showing that one pair of opposite sides are parallel and the other pair of opposite sides are not parallel. 2. We then show that the non-parallel sides (the legs) are congruent.

Example 2 A quadrilateral WXYZ four of its vertices given as W 1, 3 , X −1, 1 , Y −1, −2 and Z 4, 3 . Show that this quadrilateral is an isosceles trapezoid. Solution We first show that the quadrilateral is a trapezoid by finding the slopes of all its sides then we show that the non-parallel sides are congruent. Slope of WX= 3−1 1+1 = 2 2 =1 Slope of XY= −2−1 −1+1 = −3 0 =𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑

Slope of YZ= 3+2 4+1 = 5 5 =1 Slope of WZ= 3−3 4−1 = 0 3 =0 The slopes show that 𝑊𝑋∥𝑌𝑍 and 𝑊𝑍∦𝑋𝑌, therefore quadrilateral WXYZ is a trapezium. We then show that the non-parallel sides are congruent. The non-parallel sides are WZ and XY.

𝑊𝑍= 4−1 2 + 3−3 2 =3 units 𝑋𝑌= −1+1 2 + −2−1 2 =3 units The non parallel sides are congruent hence quadrilateral WXYZ is an isosceles trapezoid.

Determining the coordinates of the missing vertex that makes up a isosceles trapezoid In coordinate geometry, it is possible to use the properties of a given quadrilateral to find the coordinates of a missing vertex when the coordinates of the other three vertices are known. We will learn how to find the coordinates of the missing vertex that make up an isosceles trapezoid.

Example 3 CATE is an isosceles trapezoid on the x-y plane whose three vertices are given as 𝐶 4, 6 , 𝐴 11, 6 and 𝐸 6, 10 Find the coordinates of vertex T. Solution We sketch the given points on the isosceles trapezoid as shown below. Since ET is horizontal, E and T will have the same 𝑦 coordinate. C, K, L and A will also have the same y coordinate.

C and K are on the same horizontal line, therefore we can determine the coordinates of K. K will have the same 𝑦 coordinate as C and the same 𝑥 coordinate as E, hence we get the coordinates 𝐾 6, 6 𝐸 6, 10 𝐶 4, 6 𝐴 11, 6 T K L

The distance CK and AL are the same because it is an isosceles trapezoid. 𝐶𝐾=𝐴𝐿=2 𝑢𝑛𝑖𝑡𝑠 The coordinates of T can be determined by considering the coordinates of L and E. 𝐿 11−2, 6 ⇒𝐿 9, 6 T will have the same 𝑦−coordinate as E Therefore the coordinates of T will be given as: 𝑇 9, 10

homework PQRS is an isosceles trapezoid drawn in the first quadrant of the 𝑥−𝑦 plane. Three of its vertices have the coordinates: 𝑃 2, 2 , 𝑄 9,2 and 𝑅 7,5 . Determine the coordinates of vertex S.

Answers to homework S 4,5

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