Source: Looking for Pythagoras from the Connected Math Project ...

Math 103 – Background: Plotting points, linear measurements, Pythagoras
Source: Looking for Pythagoras from the Connected Math Project, grade 8
Locating points, finding slopes:
Coordinate systems are used to describe the location of points; the horizontal and vertical number lines are
called the x- and y-axis, respectively. You can describe the location of a point in the plane by an ordered
x, y . On the grid below, four points are four points, labeled with their coordinates.
pair ( )
Questions:
The coordinate system described here is called a
rectangular coordinate system. (Later, we’ll look at
another system for locating points called the polar system.
Polar coordinates consist of a distance and an angle. The
four-in-a-row game practiced this type of labeling.)
A(
−2,
1)
B ( 2,
4)
C ( 6,
− 2)
D ( −2,
− 5)
1. The slope of a line is the ratio of the vertical change to the horizontal change.
a. Determine the slope of the line that joins A to B, denoted AB; and the slope of CD. (above)
b. Perpendicular lines intersect at right angles. Find the coordinates of a point E, so that CE is
perpendicular to CD. Find three answers. (above)
c. Find the slope of the segment CE. What do you notice about the slopes of perpendicular lines?
2. Which triangles are right triangles?
Explain your reasoning!

3. The points ( 0 , 0 ) and ( , 2 )
3 are two vertices of a polygon.
a. If the polygon is square, find the other two vertices. (There are two squares that answer this
question, find both.)
b. If the polygon is a nonrectangular parallelogram, what might the other two vertices be? (Infinitely
many answers are possible.)
c. If the polygon is a right triangle, what might the other vertex be? Find two. (Infinitely many
answers are possible.)
d. The points ( 3 , 3 ) and ( , 6 )
2 are two vertices of a right triangle. List at least three points that
could be the other vertex.
4. On grid paper, draw several parallelograms with diagonals that are perpendicular. What do you
observe about these parallelograms?
Finding area and lengths:
1. Find the area of each figure below. Consider the horizontal or vertical distance between adjacent dots
to be one unit. (Curriculum addition: Describe the strategy that you used.)

2. The smallest square you can draw by connecting dots on a five-by-five dot grid is a 1-by-1 unit square,
which has area one square unit, as shown. You can also draw a 2-by-2, which has area 4 square units,
as well as other larger square. These square are called upright. You can also draw tilted squares, like
last one shown. This square has area 2. Explain why this square has area 2.
3. On the grid paper below, find as many upright and as many tilted squares that can be drawn on a fiveby-five
grid by connecting dots. Write the area of each square inside the square. Be sure you can
explain how you can be sure that each tilted square you drew really is square.

Intro to square roots: The area of a square is the length of a side multiplied by itself. This can be
2
expressed by the formula A = s × s or A = s . If you know the area of a square, you can work backwards
to find the length of a side. For example, if a square has area 4 square units, then the length of a side is 2 –
we call 2 a square root of 4 and write this 2 = 4 . If a square has area 2 square units, we call the length of
a side 2 . To find the length of a line segment which is not vertical or horizontal, we could use areas of
squares like this:
4. Express the length of the side of each tilted square that you found in problem 3 above, using
notation.
5. Express the length of the segment shown using
numbers is this length?
notation. Between what two consecutive whole
Pythagorean Theorem:
Consider a right triangle with legs as shown.
1. How are the areas of the attached squares related?
2. Construct line segments by connecting dots on the attached grid paper of the following lengths:
a. 13 b. 17 c. 20 d. 34