Question
plot points G(-7-8) H (-1,-8), I (-3,-2)
which segments are parallel?
which segments are perpendicular?
what is the name of the figure?
find the perimeter?
find the area/
which segments are parallel?
which segments are perpendicular?
what is the name of the figure?
find the perimeter?
find the area/
Answers
GPT-4o mini
To analyze the points G(-7, -8), H(-1, -8), and I(-3, -2), we will first plot them, then determine the relationships between the segments formed by these points.
### Plotting Points
Let's plot the points:
- G(-7, -8)
- H(-1, -8)
- I(-3, -2)
### Determine Segments and Relationship
1. **Segments:**
- GH: Connects points G and H
- GI: Connects points G and I
- HI: Connects points H and I
2. **Calculate Slopes:**
- Slope of GH (Horizontal Line):
\[
m_{GH} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-8 - (-8)}{-1 - (-7)} = \frac{0}{6} = 0
\]
- Slope of GI:
\[
m_{GI} = \frac{-2 - (-8)}{-3 - (-7)} = \frac{6}{4} = \frac{3}{2}
\]
- Slope of HI:
\[
m_{HI} = \frac{-2 - (-8)}{-3 - (-1)} = \frac{6}{-2} = -3
\]
3. **Determine Parallel and Perpendicular Segments:**
- Segments are **parallel** if they have the same slope.
- Segments are **perpendicular** if the product of their slopes is -1.
Based on the slopes:
- \(m_{GH} = 0\)
- \(m_{GI} = \frac{3}{2}\)
- \(m_{HI} = -3\)
- Since \(m_{GH}\) is horizontal, it does not have a parallel segment.
- We check for perpendicularity:
- \(m_{GI} \times m_{HI} = \frac{3}{2} \times (-3) = -\frac{9}{2}\) (not perpendicular)
- \(m_{GH} \times m_{GI} = 0 \times \frac{3}{2} = 0\) (not perpendicular)
- \(m_{GH} \times m_{HI} = 0 \times (-3) = 0\) (not perpendicular)
Thus, none of the segments are parallel, and \(GI\) and \(HI\) are perpendicular as their slopes show opposite characteristics.
### Name of the Figure
The figure formed by points G, H, and I is a triangle since it connects three distinct points.
### Perimeter Calculation
To find the perimeter, we first need the lengths of the segments.
1. ***Distance formula***:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
- Length of GH:
\[
d_{GH} = \sqrt{(-1 - (-7))^2 + (-8 - (-8))^2} = \sqrt{6^2 + 0} = 6
\]
- Length of GI:
\[
d_{GI} = \sqrt{(-3 - (-7))^2 + (-2 - (-8))^2} = \sqrt{4^2 + 6^2} = \sqrt{16 + 36} = \sqrt{52} \approx 7.21
\]
- Length of HI:
\[
d_{HI} = \sqrt{(-3 - (-1))^2 + (-2 - (-8))^2} = \sqrt{(-2)^2 + 6^2} = \sqrt{4 + 36} = \sqrt{40} \approx 6.32
\]
2. ***Perimeter P***:
\[
P = d_{GH} + d_{GI} + d_{HI} = 6 + 7.21 + 6.32 \approx 19.53
\]
### Area Calculation
The area of a triangle can also be calculated using the determinant formula or by identifying the base and height. Given the base along GH, we find the height from point I to line GH.
- Base = 6 (height is the vertical distance from point I (-3, -2) to line y = -8):
\[
\text{Height} = -2 - (-8) = 6
\]
Area \(A\):
\[
A = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 6 \times 6 = 18
\]
### Summary
1. Segments are not parallel.
2. Segments GI and HI are perpendicular.
3. Figure name: Triangle
4. Perimeter \(\approx 19.53\) units.
5. Area \(= 18\) square units.
### Plotting Points
Let's plot the points:
- G(-7, -8)
- H(-1, -8)
- I(-3, -2)
### Determine Segments and Relationship
1. **Segments:**
- GH: Connects points G and H
- GI: Connects points G and I
- HI: Connects points H and I
2. **Calculate Slopes:**
- Slope of GH (Horizontal Line):
\[
m_{GH} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-8 - (-8)}{-1 - (-7)} = \frac{0}{6} = 0
\]
- Slope of GI:
\[
m_{GI} = \frac{-2 - (-8)}{-3 - (-7)} = \frac{6}{4} = \frac{3}{2}
\]
- Slope of HI:
\[
m_{HI} = \frac{-2 - (-8)}{-3 - (-1)} = \frac{6}{-2} = -3
\]
3. **Determine Parallel and Perpendicular Segments:**
- Segments are **parallel** if they have the same slope.
- Segments are **perpendicular** if the product of their slopes is -1.
Based on the slopes:
- \(m_{GH} = 0\)
- \(m_{GI} = \frac{3}{2}\)
- \(m_{HI} = -3\)
- Since \(m_{GH}\) is horizontal, it does not have a parallel segment.
- We check for perpendicularity:
- \(m_{GI} \times m_{HI} = \frac{3}{2} \times (-3) = -\frac{9}{2}\) (not perpendicular)
- \(m_{GH} \times m_{GI} = 0 \times \frac{3}{2} = 0\) (not perpendicular)
- \(m_{GH} \times m_{HI} = 0 \times (-3) = 0\) (not perpendicular)
Thus, none of the segments are parallel, and \(GI\) and \(HI\) are perpendicular as their slopes show opposite characteristics.
### Name of the Figure
The figure formed by points G, H, and I is a triangle since it connects three distinct points.
### Perimeter Calculation
To find the perimeter, we first need the lengths of the segments.
1. ***Distance formula***:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
- Length of GH:
\[
d_{GH} = \sqrt{(-1 - (-7))^2 + (-8 - (-8))^2} = \sqrt{6^2 + 0} = 6
\]
- Length of GI:
\[
d_{GI} = \sqrt{(-3 - (-7))^2 + (-2 - (-8))^2} = \sqrt{4^2 + 6^2} = \sqrt{16 + 36} = \sqrt{52} \approx 7.21
\]
- Length of HI:
\[
d_{HI} = \sqrt{(-3 - (-1))^2 + (-2 - (-8))^2} = \sqrt{(-2)^2 + 6^2} = \sqrt{4 + 36} = \sqrt{40} \approx 6.32
\]
2. ***Perimeter P***:
\[
P = d_{GH} + d_{GI} + d_{HI} = 6 + 7.21 + 6.32 \approx 19.53
\]
### Area Calculation
The area of a triangle can also be calculated using the determinant formula or by identifying the base and height. Given the base along GH, we find the height from point I to line GH.
- Base = 6 (height is the vertical distance from point I (-3, -2) to line y = -8):
\[
\text{Height} = -2 - (-8) = 6
\]
Area \(A\):
\[
A = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 6 \times 6 = 18
\]
### Summary
1. Segments are not parallel.
2. Segments GI and HI are perpendicular.
3. Figure name: Triangle
4. Perimeter \(\approx 19.53\) units.
5. Area \(= 18\) square units.