Question
Mia is riding her bike from her house to the park if the park is located at (3, -5) and Mia lives 7 units away from the park what is a possible coordinate of Mia's house
Answers
GPT-4o mini
To find a possible coordinate for Mia's house, we need to determine a point that is 7 units away from the park located at (3, -5). This can be done using the distance formula:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
In this case, the distance \( d = 7 \), and the park coordinates are \( (x_2, y_2) = (3, -5) \). Let \( (x_1, y_1) \) represent the coordinates of Mia's house.
Applying the distance formula, we rearrange it for Mia's house:
\[
7 = \sqrt{(x_1 - 3)^2 + (y_1 + 5)^2}
\]
Squaring both sides, we get:
\[
49 = (x_1 - 3)^2 + (y_1 + 5)^2
\]
This equation describes a circle centered at the point (3, -5) with a radius of 7. We can find potential coordinates by choosing values for \( x_1 \) and calculating \( y_1 \).
1. Let's say we choose \( x_1 = 3 \) (the same x-coordinate as the park):
\[
49 = (3 - 3)^2 + (y_1 + 5)^2
\]
\[
49 = 0 + (y_1 + 5)^2
\]
\[
(y_1 + 5)^2 = 49
\]
\[
y_1 + 5 = 7 \quad \text{or} \quad y_1 + 5 = -7
\]
This gives:
- \( y_1 = 2 \) (one possible coordinate is \( (3, 2) \))
- \( y_1 = -12 \) (another possible coordinate is \( (3, -12) \))
2. Alternatively, let’s choose \( x_1 = 0 \):
\[
49 = (0 - 3)^2 + (y_1 + 5)^2
\]
\[
49 = 9 + (y_1 + 5)^2
\]
\[
40 = (y_1 + 5)^2
\]
\[
y_1 + 5 = \sqrt{40} \quad \text{or} \quad y_1 + 5 = -\sqrt{40}
\]
This gives:
\[
y_1 \approx 2.32 \quad \text{or} \quad y_1 \approx -12.32
\]
So we could also have \( (0, 2.32) \) or \( (0, -12.32) \).
Thus, some possible coordinates for Mia's house include \( (3, 2) \), \( (3, -12) \), \( (0, 2.32) \), and \( (0, -12.32) \).
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
In this case, the distance \( d = 7 \), and the park coordinates are \( (x_2, y_2) = (3, -5) \). Let \( (x_1, y_1) \) represent the coordinates of Mia's house.
Applying the distance formula, we rearrange it for Mia's house:
\[
7 = \sqrt{(x_1 - 3)^2 + (y_1 + 5)^2}
\]
Squaring both sides, we get:
\[
49 = (x_1 - 3)^2 + (y_1 + 5)^2
\]
This equation describes a circle centered at the point (3, -5) with a radius of 7. We can find potential coordinates by choosing values for \( x_1 \) and calculating \( y_1 \).
1. Let's say we choose \( x_1 = 3 \) (the same x-coordinate as the park):
\[
49 = (3 - 3)^2 + (y_1 + 5)^2
\]
\[
49 = 0 + (y_1 + 5)^2
\]
\[
(y_1 + 5)^2 = 49
\]
\[
y_1 + 5 = 7 \quad \text{or} \quad y_1 + 5 = -7
\]
This gives:
- \( y_1 = 2 \) (one possible coordinate is \( (3, 2) \))
- \( y_1 = -12 \) (another possible coordinate is \( (3, -12) \))
2. Alternatively, let’s choose \( x_1 = 0 \):
\[
49 = (0 - 3)^2 + (y_1 + 5)^2
\]
\[
49 = 9 + (y_1 + 5)^2
\]
\[
40 = (y_1 + 5)^2
\]
\[
y_1 + 5 = \sqrt{40} \quad \text{or} \quad y_1 + 5 = -\sqrt{40}
\]
This gives:
\[
y_1 \approx 2.32 \quad \text{or} \quad y_1 \approx -12.32
\]
So we could also have \( (0, 2.32) \) or \( (0, -12.32) \).
Thus, some possible coordinates for Mia's house include \( (3, 2) \), \( (3, -12) \), \( (0, 2.32) \), and \( (0, -12.32) \).