Lesson 10

Parallel Lines in the Plane

  • Let’s investigate parallel lines in the coordinate plane.

Problem 1

Select all equations that are parallel to the line \(2x + 5y = 8\).

A:

\(y=\frac{2}{5}x + 4\)

B:

\(y=\text-\frac{2}{5}x + 4\)

C:

\(y-2=\frac{5}{2}(x+1)\)

D:

\(y-2=\text-\frac{2}{5}(x+1)\)

E:

\(10x+5y=40\)

Problem 2

Prove that \(ABCD\) is not a parallelogram.

Quadrilateral ABCD. A at 1 comma 1, B at 4 comma 6, C at 9 comma 6, D at 5 comma 1.

Problem 3

Write an equation of a line that passes through \((\text-1,2)\) and is parallel to a line with \(x\)-intercept \((3,0)\) and \(y\)-intercept \((0,1)\). 

Problem 4

Write an equation of the line with slope \(\frac23\) that goes through the point \((\text-2, 5)\).

(From Unit 6, Lesson 9.)

Problem 5

Priya and Han each wrote an equation of a line with slope \(\frac13\) that passes through the point \((1,2)\). Priya’s equation is \(y-2 = \frac13 (x-1)\) and Han’s equation is \(3y - x = 5\). Do you agree with either of them? Explain or show your reasoning. 

(From Unit 6, Lesson 9.)

Problem 6

Match each equation with another equation whose graph is the same parabola.

A:

\((x-3)^2+(y-2)^2=y^2\)

B:

\((x-2)^2+(y-3)^2=(y+3)^2\)

C:

\((x-3)^2+(y-4)^2=(y+2)^2\)

D:

\((x-2)^2+(y-2)^2=(y+2)^2\)

1:

\(y=\frac18(x-2)^2\)

2:

\(y=\frac{1}{12}(x-2)^2\)

3:

\(y=\frac{1}{4}(x-3)^2+1\)

4:

\(y=\frac{1}{12}(x-3)^2+1\)

(From Unit 6, Lesson 8.)

Problem 7

A parabola is defined as the set of points the same distance from \((\text-1, 3)\) and the line \(y=5\). Select the point that is on this parabola.

A:

\((\text-1, 3)\)

B:

\((0, 5)\)

C:

\((3,0)\)

D:

\((0,0)\)

(From Unit 6, Lesson 7.)

Problem 8

Here are some transformation rules. For each rule, describe whether the transformation is a rigid motion, a dilation, or neither.

  1. \((x,y) \rightarrow (2x,y+2)\)
  2. \((x,y) \rightarrow (2x,2y)\)
  3. \((x,y) \rightarrow (x+2,y+2)\)
  4. \((x,y) \rightarrow (x-2,y)\)
(From Unit 6, Lesson 2.)