Exercise 54 Page 195

a We can start with plotting points

A (2, - 4)

and

C (10, 4)

on a coordinate plane. Let's recall that a square is a quadrilateral with four right angles and four equal side lengths. To find the last two points of the square, we can use the vertical and horizontal lines of the coordinate plane and draw the lines that will contain the points and the sides of the square.

Using the

x -

and

y -

axes, we can find the unknown coordinates of the points.

B (2, 4) and D (10, - 4)

Notice that the lengths of all the sides are equal and the interior angles are right angles (you can find the proof in Part C).

b Let's recall what Postulate

3.2

from the book states.

Two nonvertical lines have the same slope if and only if they are parallel . All vertical lines are parallel .

Since the segments

\overline{A B}

and

\overline{C D}

lie on the vertical lines, they are parallel. Now, we can calculate the slopes of the horizontal segments and check whether they have the same slopes.

Slope of $\overline{A D}$

In order to calculate the slope of the line, we can substitute two points on the line into the Slope Formula. Let's substitute

A (2, - 4)

and

D (10, - 4) .

m_{1} = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}

SubstitutePoints

Substitute $(2, - 4)$ & $(10, - 4)$

m_{1} = \frac{- 4 - ( - 4 )}{1 0 - 2}

SubNeg

$a - (- b) = a + b$

m_{1} = \frac{- 4 + 4}{1 0 - 2}

AddSubTerms

Add and subtract terms

m_{1} = \frac{0}{8}

CalcQuot

Calculate quotient

m_{1} = 0

The slope of the line and respectively the segment

\overline{A D}

0 .

Slope of $\overline{B C}$

Similarly, by substituting the points

B (2, 4)

and

C (10, 4)

into the Slope Formula, we can calculate the slope of the line.

m_{2} = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}

SubstitutePoints

Substitute $(2, 4)$ & $(10, 4)$

m_{2} = \frac{4 - 4}{1 0 - 2}

SubTerms

Subtract terms

m_{2} = \frac{0}{8}

CalcQuot

Calculate quotient

m_{2} = 0

Thus, the slope of this line and segment

\overline{B C}

is also

0 .

Conclusion

We calculated the slopes of both segments.

m_{1} = 0 m_{2} = 0

Since they are the same, we conclude that segments

\overline{A D}

and

\overline{B C}

are parallel.

c From Postulate

3.3

we know that vertical and horizontal lines are perpendicular— they form right, or

9 0^{\circ}

angles. In Part A, to find the coordinates of the unknown points we used

2

vertical and

2

horizontal lines. These lines contain the sides of the square, which means that they are also perpendicular.

Segments $A D$ and $A B$ lie exactly on one horizontal and one vertical line respectively. The angle they form is a right angle. We can analyze the rest of the angles the same way and get the same result.

Subchapter links

Check Your Understanding p.192

Practice and Problem Solving p.193-195

H.O.T. Problems p.195

Standardized Test Practice p.196

Spiral Review p.196

Skills Review p.196