Distance, Midpoint, Coordinate Geometry and Quadrilaterals
Activity: Participants are placed in groups of three or four and are given these questions. After each,
class discussions should be held in order to gain a framework for the problem solving strategies that
were used as well as the properties of the quadrilateral that were needed to answer each question.
Given that the coordinates for the endpoints of the diagonal of a square are at (2,1) and (5,4), find the
other two vertices of the square.
Find the coordinates for the other two vertices of a square that has a diagonal with endpoints at (2,1) and
(5,1).
Find the other two vertices of a square if the endpoint of a side are at (-4,-2) and (-1,-1).
Find the other two vertices of a rectangle if the endpoint of a side are at (-4,-2) and (-1,-1) and the other
side measurement is 2√10.
Find the other two vertices of a square when given a diagonal with endpoints at (2,1) and (5,3).
Find the other two vertices of a square when given a diagonal with endpoints at (-a,b) and (a,b).
Find two coordinates that will complete the non-square rhombus that has a diagonal with endpoints at
(2,1) and (5,1).
Find two coordinates that will complete the rhombus that has a diagonal with endpoints at (2,1) and
(5,3).
 What properties of rhombuses are you using for this problem and what prerequisite knowledge is
necessary/useful?
 Find the equation that represents every possible coordinate. How did you find it?
Find two coordinates that will complete a rectangle that has a diagonal with endpoints at (-5,4) and (3,2).
 Find integer coordinates.
 Find the coordinates that make it a square.
 What properties of rectangles are you using for this problem and what prerequisite knowledge is
necessary/useful?
 Find the equation that represents every possible coordinate. How did you find it?
Distance, Midpoint, Coordinate Geometry and Quadrilaterals
Mathematical Goals: Teachers will be able to
 Determine the remaining vertices of a quadrilateral on the coordinate plane when given two of the
coordinates.
 Determine which properties of quadrilaterals were used to calculate these vertices.
Pedagogical Goals: Teachers will be able to
 Determine which of the Standards of Mathematical Practice are addressed in this activity.
 Provide appropriate modifications to this activity for use in their classrooms.
 Gain an understanding of students’ struggles with coordinate geometry through the use of highlevel problems.
Technological Goals:
 None
Alignment to the Common Core:
 G-GPE.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or
disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or
disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).
 G-C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles
Mathematical Practices:
1 – Make sense of problems and persevere in solving them
3 – Construct viable arguments and critique the reasoning of others
4 – Model with mathematics
5 – Use appropriate tools strategically
8 – Look for and express regularity in repeated reasoning
Length of Lesson:
90 minutes
Materials Needed:
Graph paper
Overview:
Participants will explore the properties of quadrilaterals using coordinate geometry with a series of
increasingly difficult questions. Each question will give the coordinates of the endpoints of a diagonal of a
specified quadrilateral and will ask for the other two vertices of the quadrilateral.
Activity: Participants are placed in groups of three or four and are given these questions. After each,
class discussions should be held in order to gain a framework for the problem solving strategies that
were used as well as the properties of the quadrilateral that were needed to answer each question.
Given that the coordinates for the endpoints of the diagonal of a square are at (2,1) and (5,4), find the
other two vertices of the square.
(5,1) and (2,4)
Find the coordinates for the other two vertices of a square that has a diagonal with endpoints at (2,1) and
(5,1).
(3.5,2.5) and (3.5, -1.5)
Pedagogical discussion: What properties of the square did you use? What background knowledge
is necessary for this calculation?
Find the other two vertices of a square if the endpoint of a side are at (-4,-2) and (-1,-1).
There is more than one set of solutions for this, and this can be found using reciprocal slopes, thus
exploring the properties of the sides of a square more so than the properties of the diagonals. (5,1) and (-2,2) or (-3,-5) and (0,-4)
Find the other two vertices of a rectangle if the endpoint of a side are at (-4,-2) and (-1,-1) and the other
side measurement is 2√10.
As with the previous problem there is more than one set of solutions for this. In this case, it is
necessary to determine the distance between the two given coordinates and use the negative
reciprocals of the slope.
(-6,4) and (-3,5) or (-2,-8) and (1,-7)
Find the other two vertices of a square when given a diagonal with endpoints at (2,1) and (5,3).
There are multiple entry points for this problem. One is to find the perpendicular bisector of the
diagonal and determine the endpoints of this diagonal. They are (2,3.5) and (4.5,0.5).
Find the other two vertices of a square when given a diagonal with endpoints at (-a,b) and (a,b).
(0, a+b) and (0, b-a)
Find two coordinates that will complete the non-square rhombus that has a diagonal with endpoints at
(2,1) and (5,1).
(Why was it necessary to say non-square?)
(3.5, 1±n)
Find two coordinates that will complete the rhombus that has a diagonal with endpoints at (2,1) and
(5,3).
 What properties of rhombuses are you using for this problem and what prerequisite knowledge is
necessary/useful?
The perpendicularity of the diagonals is the most useful property of rhombuses.
 Find the equation that represents every possible coordinate. How did you find it?
The equation that represents every possible coordinate is a line. The slope between the given
coordinates is 2/3, so the slope of the line is -3/2. The equation of this line is y=-3/2 x +7.25.
Find two coordinates that will complete a rectangle that has a diagonal with endpoints at (-5,4) and (3,2).
 Find integer coordinates.
Any of the following pairs of coordinates are correct



(-4,5) and (2,-3)
(-1,6) and (-1,4)
(2,5) and (-4,-3)
(4,4) and (-5,-2)
(4,1) and (-6,1)
(-5,-2) and (3,4)
Find the coordinates that make it a square.
(2,5) and (-4,-3)
What properties of rectangles are you using for this problem and what prerequisite knowledge is
necessary/useful?
The diagonals of a rectangle are congruent, and this makes the given diagonal of the rectangle the
diameter of a circle with radius 5.
Find the equation that represents every possible coordinate. How did you find it? (The solution is
the circle (x+1)2+(y+1)2=25)