given a(1, -3) and b(-7, -7) are on the same line.
write the equation for the line that is parallel to line ab that goes through c(7, 1).
a. slope of ab:
b. equation of $\overleftrightarrow{ab}$:
(old line)
c. equation of line parallel through c:
(new line)
10. graph the line.
11. graph the line.
given a(5, -8) and b(0, -9) are on the same line.
write the equation for the line that is perpendicular to line ab that goes through c(-8, 2).
a. slope of ab:
b. equation of $\overleftrightarrow{ab}$:
(old line)
c. equation of line perpendicular through c:
(new line)
12. graph the line.
13. graph the line.
classifying quadrilaterals
find the 4 - sided shape on your graph that contains the 4*
using that shape, determine if this quadrilateral fits the following definitions of special quadrilaterals:
- parallelogram – a 4 - sided shape with both pairs of opposite sides parallel
- rectangle – a 4 - sided shape with each pair of adjacent sides perpendicular
- trapezoid – a 4 - sided shape with exactly one pair of opposite sides parallel
12. is the shape with 4* a parallelogram, a rectangle, or a trapezoid?
state the type of special quadrilateral and provide evidence. if not, provide evidence.

Question

Accepted Answer

### Part 1: First Line (A(1, -3) and B(-7, -7))
#### a. Slope of AB
# Explanation:
## Step1: Recall slope formula
The slope $ m $ between two points $ (x_1, y_1) $ and $ (x_2, y_2) $ is $ m = \frac{y_2 - y_1}{x_2 - x_1} $.
For $ A(1, -3) $ and $ B(-7, -7) $, $ x_1 = 1, y_1 = -3, x_2 = -7, y_2 = -7 $.
## Step2: Calculate slope
$ m = \frac{-7 - (-3)}{-7 - 1} = \frac{-7 + 3}{-8} = \frac{-4}{-8} = \frac{1}{2} $.
# Answer: $ \frac{1}{2} $

#### b. Equation of $ \overleftrightarrow{AB} $
# Explanation:
## Step1: Use point-slope form
Point-slope: $ y - y_1 = m(x - x_1) $. Use $ A(1, -3) $ and $ m = \frac{1}{2} $.
## Step2: Substitute values
$ y - (-3) = \frac{1}{2}(x - 1) $ → $ y + 3 = \frac{1}{2}x - \frac{1}{2} $.
## Step3: Simplify to slope-intercept
$ y = \frac{1}{2}x - \frac{1}{2} - 3 $ → $ y = \frac{1}{2}x - \frac{7}{2} $.
# Answer: $ y = \frac{1}{2}x - \frac{7}{2} $ (or $ x - 2y = 7 $)

#### c. Equation of Parallel Line through $ C(7, 1) $
# Explanation:
## Step1: Parallel lines have equal slopes
Slope of new line $ m = \frac{1}{2} $ (same as AB).
## Step2: Use point-slope with $ C(7, 1) $
$ y - 1 = \frac{1}{2}(x - 7) $.
## Step3: Simplify
$ y - 1 = \frac{1}{2}x - \frac{7}{2} $ → $ y = \frac{1}{2}x - \frac{7}{2} + 1 $ → $ y = \frac{1}{2}x - \frac{5}{2} $.
# Answer: $ y = \frac{1}{2}x - \frac{5}{2} $ (or $ x - 2y = 5 $)

### Part 2: Second Line (A(5, -8) and B(0, -9))
#### a. Slope of AB
# Explanation:
## Step1: Apply slope formula
For $ A(5, -8) $ and $ B(0, -9) $, $ x_1 = 5, y_1 = -8, x_2 = 0, y_2 = -9 $.
$ m = \frac{-9 - (-8)}{0 - 5} = \frac{-1}{-5} = \frac{1}{5} $.
# Answer: $ \frac{1}{5} $

#### b. Equation of $ \overleftrightarrow{AB} $
# Explanation:
## Step1: Point-slope with $ A(5, -8) $ and $ m = \frac{1}{5} $
$ y - (-8) = \frac{1}{5}(x - 5) $ → $ y + 8 = \frac{1}{5}x - 1 $.
## Step2: Simplify
$ y = \frac{1}{5}x - 1 - 8 $ → $ y = \frac{1}{5}x - 9 $.
# Answer: $ y = \frac{1}{5}x - 9 $ (or $ x - 5y = 45 $)

#### c. Equation of Perpendicular Line through $ C(-8, 2) $
# Explanation:
## Step1: Perpendicular slope
Perpendicular slope $ m_{\perp} = -\frac{1}{m} = -5 $ (since $ m = \frac{1}{5} $).
## Step2: Point-slope with $ C(-8, 2) $
$ y - 2 = -5(x + 8) $.
## Step3: Simplify
$ y - 2 = -5x - 40 $ → $ y = -5x - 38 $.
# Answer: $ y = -5x - 38 $ (or $ 5x + y = -38 $)

### Part 3: Classifying Quadrilaterals (Question 12)
# Brief Explanations:
To classify, we analyze side slopes (parallel/perpendicular). A parallelogram has both pairs of opposite sides parallel; a rectangle has adjacent sides perpendicular (so also a parallelogram); a trapezoid has exactly one pair parallel.
1. Find the 4-sided shape with $ 4^* $ on the graph.
2. Calculate slopes of all four sides.
3. Check:
   - **Parallelogram**: Both pairs of opposite sides have equal slopes (parallel).
   - **Rectangle**: Adjacent sides have slopes that are negative reciprocals (perpendicular), and opposite sides parallel.
   - **Trapezoid**: Exactly one pair of opposite sides has equal slopes (parallel), the other pair does not.

(Note: Since the graph is not provided, assume typical cases. For example, if two pairs of sides are parallel, it’s a parallelogram; if one pair, a trapezoid; if parallelogram with perpendicular sides, a rectangle.)

# Answer: (Example) If two pairs of opposite sides are parallel → **Parallelogram** (evidence: slopes of opposite sides are equal). If one pair → **Trapezoid**. If parallelogram with perpendicular adjacent sides → **Rectangle**. (Depends on graph slopes.)

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QUESTION IMAGE

Question

Explanation:

Part 1: First Line (A(1, -3) and B(-7, -7))

a. Slope of AB

Step1: Recall slope formula

Step2: Calculate slope

Step1: Use point-slope form

Step2: Substitute values

Step3: Simplify to slope-intercept

Step1: Parallel lines have equal slopes

Step2: Use point-slope with \( C(7, 1) \)

Step3: Simplify

Answer:

b. Equation of \( \overleftrightarrow{AB} \)