Let's carefully analyze the problem and verify the solution step by step.
We are given four lines:
- $ AB $
- $ AB_1 $
- $ A_1B $
- $ A_1B_1 $
We need to
find which line has the greatest slope.
We're given coordinates (implied from the calculations):
Let’s deduce the coordinates of the points based on the slope calculations:
---
Step 1: Identify the coordinates
From the slope formulas, we can infer:
- For
slope of AB:
$$
\frac{3 - 2}{5 - 0} = \frac{1}{5}
$$
So point $ A = (0, 2) $, $ B = (5, 3) $
- For
slope of $ AB_1 $:
$$
\frac{-3 - 2}{5 - 0} = \frac{-5}{5} = -1
$$
So $ B_1 = (5, -3) $
- For
slope of $ A_1B $:
$$
\frac{3 - (-2)}{5 - 0} = \frac{5}{5} = 1
$$
So $ A_1 = (0, -2) $
- For
slope of $ A_1B_1 $:
$$
\frac{-3 - (-2)}{5 - 0} = \frac{-1}{5} = -\frac{1}{5}
$$
So now we have:
- $ A = (0, 2) $
- $ B = (5, 3) $
- $ A_1 = (0, -2) $
- $ B_1 = (5, -3) $
Now compute all slopes again for clarity.
---
Step 2: Compute Slopes
1.
Slope of $ AB $:
$$
\frac{3 - 2}{5 - 0} = \frac{1}{5} = 0.2
$$
2.
Slope of $ AB_1 $:
$$
\frac{-3 - 2}{5 - 0} = \frac{-5}{5} = -1
$$
3.
Slope of $ A_1B $:
$$
\frac{3 - (-2)}{5 - 0} = \frac{5}{5} = 1
$$
4.
Slope of $ A_1B_1 $:
$$
\frac{-3 - (-2)}{5 - 0} = \frac{-1}{5} = -0.2
$$
---
Step 3: Compare the slopes
- $ AB $: $ 0.2 $
- $ AB_1 $: $ -1 $
- $ A_1B $: $ 1 $
- $ A_1B_1 $: $ -0.2 $
The
greatest slope is $ 1 $, which belongs to line $ A_1B $.
---
✔ Final Answer:
$$
\boxed{A_1B}
$$
This line has the
greatest slope among the four.
---
Explanation:
- Slope measures how steep a line is.
- Positive slopes go upward from left to right; negative slopes go downward.
- Among the positive slopes ($ 0.2 $ and $ 1 $), $ 1 $ is larger.
- Therefore, $ A_1B $, with slope $ 1 $, is the steepest upward-sloping line.
✔ The conclusion in your image is
correct: the greatest slope is indeed
line $ A_1B $, with slope $ 1 $.
Parent Tip: Review the logic above to help your child master the concept of sat math worksheet.
