Question

3 - 67. the diagram at right represents only half of a shape that has the graph of y = 1 as a line of symmetry. draw the completed shape on your paper, and label the coordinates of the missing vertices. 3 - 68. the length of a side of a square is 5x + 2 units. assuming the perimeter is 48 units, complete the following problems. a. write an equation to represent this information. b. solve for x. c. what is the area of the square?

Explanation:

3 - 67:

This problem involves reflecting the given half - shape over the line \(y = 1\). To find the coordinates of the missing vertices, we use the rule for reflection over a horizontal line \(y = k\). If a point \((x,y)\) is reflected over the line \(y=k\), the new point is \((x,2k - y)\). Here \(k = 1\).

3 - 68:

a.

The perimeter \(P\) of a square is given by \(P=4s\), where \(s\) is the side - length. Given \(s = 5x+2\) and \(P = 48\), the equation is \(4(5x + 2)=48\).

b.

Step1: Expand the left - hand side

\[4(5x + 2)=20x+8\]
So the equation becomes \(20x + 8=48\).

Step2: Subtract 8 from both sides

\[20x+8 - 8=48 - 8\]
\[20x=40\]

Step3: Divide both sides by 20

\[\frac{20x}{20}=\frac{40}{20}\]
\[x = 2\]

c.

First, find the side - length of the square. Substitute \(x = 2\) into \(s=5x + 2\).
\[s=5\times2+2=10 + 2=12\]
The area \(A\) of a square is \(A=s^{2}\), so \(A = 12^{2}=144\) square units.

Answer:

3 - 67:

The process of drawing the completed shape involves reflecting each vertex of the given half - shape over the line \(y = 1\) using the rule \((x,2\times1 - y)=(x,2 - y)\) for each vertex \((x,y)\) of the given half - shape.

3 - 68:

a. \(4(5x + 2)=48\)

b. \(x = 2\)

c. 144 square units