Question
Find the distance of line E. (1,5) and (9,8)
Answers
GPT-4o mini
To find the distance between two points \(E(1, 5)\) and \(F(9, 8)\) in a Cartesian plane, we can use the distance formula, which is given by:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Where \((x_1, y_1)\) is the first point and \((x_2, y_2)\) is the second point. In this case, we have:
- Point \(E(1, 5)\): \(x_1 = 1, y_1 = 5\)
- Point \(F(9, 8)\): \(x_2 = 9, y_2 = 8\)
Now we can plug these values into the formula:
\[
d = \sqrt{(9 - 1)^2 + (8 - 5)^2}
\]
Calculating the differences:
\[
d = \sqrt{(8)^2 + (3)^2}
\]
Now calculating the squares:
\[
d = \sqrt{64 + 9}
\]
Adding the results:
\[
d = \sqrt{73}
\]
Thus, the distance between the points \(E(1, 5)\) and \(F(9, 8)\) is:
\[
\sqrt{73} \approx 8.54
\]
So, the distance is \(\sqrt{73}\) or approximately 8.54 units.
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Where \((x_1, y_1)\) is the first point and \((x_2, y_2)\) is the second point. In this case, we have:
- Point \(E(1, 5)\): \(x_1 = 1, y_1 = 5\)
- Point \(F(9, 8)\): \(x_2 = 9, y_2 = 8\)
Now we can plug these values into the formula:
\[
d = \sqrt{(9 - 1)^2 + (8 - 5)^2}
\]
Calculating the differences:
\[
d = \sqrt{(8)^2 + (3)^2}
\]
Now calculating the squares:
\[
d = \sqrt{64 + 9}
\]
Adding the results:
\[
d = \sqrt{73}
\]
Thus, the distance between the points \(E(1, 5)\) and \(F(9, 8)\) is:
\[
\sqrt{73} \approx 8.54
\]
So, the distance is \(\sqrt{73}\) or approximately 8.54 units.
What about using phthagorean theorem?
GPT-4o mini
You can definitely use the Pythagorean theorem to find the distance between the two points \(E(1, 5)\) and \(F(9, 8)\), as this distance represents the hypotenuse of a right triangle formed by the horizontal and vertical distances between the two points.
1. **Determine the horizontal and vertical distances:**
- **Horizontal distance (Δx)**: This is the difference in the x-coordinates:
\[
\Delta x = x_2 - x_1 = 9 - 1 = 8
\]
- **Vertical distance (Δy)**: This is the difference in the y-coordinates:
\[
\Delta y = y_2 - y_1 = 8 - 5 = 3
\]
2. **Apply the Pythagorean theorem:**
The Pythagorean theorem states that for a right triangle:
\[
c^2 = a^2 + b^2
\]
where \(c\) is the length of the hypotenuse (the distance we are looking for), and \(a\) and \(b\) are the lengths of the other two sides (the horizontal and vertical distances in this case).
Here, we have:
- \(a = \Delta x = 8\)
- \(b = \Delta y = 3\)
Plugging in the values:
\[
c^2 = 8^2 + 3^2 = 64 + 9 = 73
\]
3. **Solve for \(c\)**:
\[
c = \sqrt{73}
\]
Thus, using the Pythagorean theorem, we confirm that the distance between the points \(E(1, 5)\) and \(F(9, 8)\) is indeed \(\sqrt{73}\) or approximately \(8.54\) units, just as we found using the distance formula.
1. **Determine the horizontal and vertical distances:**
- **Horizontal distance (Δx)**: This is the difference in the x-coordinates:
\[
\Delta x = x_2 - x_1 = 9 - 1 = 8
\]
- **Vertical distance (Δy)**: This is the difference in the y-coordinates:
\[
\Delta y = y_2 - y_1 = 8 - 5 = 3
\]
2. **Apply the Pythagorean theorem:**
The Pythagorean theorem states that for a right triangle:
\[
c^2 = a^2 + b^2
\]
where \(c\) is the length of the hypotenuse (the distance we are looking for), and \(a\) and \(b\) are the lengths of the other two sides (the horizontal and vertical distances in this case).
Here, we have:
- \(a = \Delta x = 8\)
- \(b = \Delta y = 3\)
Plugging in the values:
\[
c^2 = 8^2 + 3^2 = 64 + 9 = 73
\]
3. **Solve for \(c\)**:
\[
c = \sqrt{73}
\]
Thus, using the Pythagorean theorem, we confirm that the distance between the points \(E(1, 5)\) and \(F(9, 8)\) is indeed \(\sqrt{73}\) or approximately \(8.54\) units, just as we found using the distance formula.