onsider the points $$ A(-7,10), B(12,7), C(10,-24) $$, and $$ D(-8,-3) $$. Which two aes determined by these points are perpendicular? Explain.

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To determine which two lines determined by the points $$ A(-7,10) $$, $$ B(12,7) $$, $$ C(10,-24) $$, and $$ D(-8,-3) $$ are perpendicular, we need to calculate the slopes of all possible line segments and identify any pair with slopes that are negative reciprocals of each other (since perpendicular lines have slopes $$ m_1 $$ and $$ m_2 $$ such that $$ m_1 \times m_2 = -1 $$).

### Step 1: Calculate the Slopes

1. **Slope of $$ \overline{AB} $$**:
   $$
   m_{AB} = \frac{7 - 10}{12 - (-7)} = \frac{-3}{19}
   $$

2. **Slope of $$ \overline{AC} $$**:
   $$
   m_{AC} = \frac{-24 - 10}{10 - (-7)} = \frac{-34}{17} = -2
   $$

3. **Slope of $$ \overline{AD} $$**:
   $$
   m_{AD} = \frac{-3 - 10}{-8 - (-7)} = \frac{-13}{-1} = 13
   $$

4. **Slope of $$ \overline{BC} $$**:
   $$
   m_{BC} = \frac{-24 - 7}{10 - 12} = \frac{-31}{-2} = 15.5
   $$

5. **Slope of $$ \overline{BD} $$**:
   $$
   m_{BD} = \frac{-3 - 7}{-8 - 12} = \frac{-10}{-20} = 0.5 \quad (\text{which is } \frac{1}{2})
   $$

6. **Slope of $$ \overline{CD} $$**:
   $$
   m_{CD} = \frac{-3 - (-24)}{-8 - 10} = \frac{21}{-18} = -\frac{7}{6}
   $$

### Step 2: Identify Perpendicular Slopes

To find perpendicular lines, we look for slope pairs where one slope is the negative reciprocal of the other:
- $$ m_{AC} = -2 $$
- $$ m_{BD} = 0.5 = \frac{1}{2} $$

Notice that:
$$
-2 \times \frac{1}{2} = -1
$$
This satisfies the condition for perpendicularity.

### Conclusion

**The lines $$ \overline{AC} $$ and $$ \overline{BD} $$ are perpendicular** because their slopes are $$ -2 $$ and $$ \frac{1}{2} $$, respectively, which are negative reciprocals of each other.

**Answer:**  
The lines AC and BD are perpendicular because their slopes –2 and ½ are negative reciprocals.
The lines AC and BD are perpendicular because their slopes are negative reciprocals of each other.

onsider the points , and . Which two
aes determined by these points are perpendicular? Explain.

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onsider the points , and . Which two aes determined by these points are perpendicular? Explain.