Question
Emanuel would like to build a ramp for her dog over some stairs
1. Compute the distance between points B and C
1. Compute the distance between points B and C
Answers
Answer
Y. 12 11 10 9 8 7 6 5 4 3 2 1 0 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 -11 -12
X. 12 11 10 9 8 7 6 5 4 3 2 1 0 -1- 2 -3 -4 -5 -6 -7 -8 -9 -10 -11 -12
A(-7,6)
B(-2,-6)
C(10,-1)
D(5,11)
X. 12 11 10 9 8 7 6 5 4 3 2 1 0 -1- 2 -3 -4 -5 -6 -7 -8 -9 -10 -11 -12
A(-7,6)
B(-2,-6)
C(10,-1)
D(5,11)
Answer
2. Compete the distance between point A and B.
Answer
Compute the distance between points D and B.
Answer
Use the Pythagorean theorem to compute the distance between points D and C.
Answer
What are those triangle things
Answered by
GPT-4o mini
To help you compute the distance between points B and C, I would need the coordinates of points B and C (for example, in terms of x and y coordinates) or any information on the positions of the points relative to each other. Could you please provide that information?
Answered by
GPT-4o mini
To find the distance between points B and C, we can use the distance formula:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
where \((x_1, y_1)\) are the coordinates of point B and \((x_2, y_2)\) are the coordinates of point C.
Given:
- \(B(-2, -6)\) with \(x_1 = -2\) and \(y_1 = -6\)
- \(C(10, -1)\) with \(x_2 = 10\) and \(y_2 = -1\)
Now plug in the values:
\[
d = \sqrt{(10 - (-2))^2 + (-1 - (-6))^2}
\]
\[
= \sqrt{(10 + 2)^2 + (-1 + 6)^2}
\]
\[
= \sqrt{(12)^2 + (5)^2}
\]
\[
= \sqrt{144 + 25}
\]
\[
= \sqrt{169}
\]
\[
= 13
\]
Thus, the distance between points B and C is **13 units**.
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
where \((x_1, y_1)\) are the coordinates of point B and \((x_2, y_2)\) are the coordinates of point C.
Given:
- \(B(-2, -6)\) with \(x_1 = -2\) and \(y_1 = -6\)
- \(C(10, -1)\) with \(x_2 = 10\) and \(y_2 = -1\)
Now plug in the values:
\[
d = \sqrt{(10 - (-2))^2 + (-1 - (-6))^2}
\]
\[
= \sqrt{(10 + 2)^2 + (-1 + 6)^2}
\]
\[
= \sqrt{(12)^2 + (5)^2}
\]
\[
= \sqrt{144 + 25}
\]
\[
= \sqrt{169}
\]
\[
= 13
\]
Thus, the distance between points B and C is **13 units**.
Answered by
GPT-4o mini
To find the distance between points A and B, we will again use the distance formula:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Given:
- \(A(-7, 6)\) where \(x_1 = -7\) and \(y_1 = 6\)
- \(B(-2, -6)\) where \(x_2 = -2\) and \(y_2 = -6\)
Now plug in the values:
\[
d = \sqrt{(-2 - (-7))^2 + (-6 - 6)^2}
\]
\[
= \sqrt{(-2 + 7)^2 + (-6 - 6)^2}
\]
\[
= \sqrt{(5)^2 + (-12)^2}
\]
\[
= \sqrt{25 + 144}
\]
\[
= \sqrt{169}
\]
\[
= 13
\]
Thus, the distance between points A and B is **13 units**.
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Given:
- \(A(-7, 6)\) where \(x_1 = -7\) and \(y_1 = 6\)
- \(B(-2, -6)\) where \(x_2 = -2\) and \(y_2 = -6\)
Now plug in the values:
\[
d = \sqrt{(-2 - (-7))^2 + (-6 - 6)^2}
\]
\[
= \sqrt{(-2 + 7)^2 + (-6 - 6)^2}
\]
\[
= \sqrt{(5)^2 + (-12)^2}
\]
\[
= \sqrt{25 + 144}
\]
\[
= \sqrt{169}
\]
\[
= 13
\]
Thus, the distance between points A and B is **13 units**.
Answered by
GPT-4o mini
To find the distance between points D and B, we will again use the distance formula:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Given:
- \(D(5, 11)\) where \(x_1 = 5\) and \(y_1 = 11\)
- \(B(-2, -6)\) where \(x_2 = -2\) and \(y_2 = -6\)
Now plug in the values:
\[
d = \sqrt{(-2 - 5)^2 + (-6 - 11)^2}
\]
\[
= \sqrt{(-7)^2 + (-17)^2}
\]
\[
= \sqrt{49 + 289}
\]
\[
= \sqrt{338}
\]
The square root of 338 does not simplify further, but we can approximate if necessary:
\[
\sqrt{338} \approx 18.38
\]
Thus, the distance between points D and B is approximately **18.38 units**. If you prefer the exact form, it remains \(\sqrt{338}\).
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Given:
- \(D(5, 11)\) where \(x_1 = 5\) and \(y_1 = 11\)
- \(B(-2, -6)\) where \(x_2 = -2\) and \(y_2 = -6\)
Now plug in the values:
\[
d = \sqrt{(-2 - 5)^2 + (-6 - 11)^2}
\]
\[
= \sqrt{(-7)^2 + (-17)^2}
\]
\[
= \sqrt{49 + 289}
\]
\[
= \sqrt{338}
\]
The square root of 338 does not simplify further, but we can approximate if necessary:
\[
\sqrt{338} \approx 18.38
\]
Thus, the distance between points D and B is approximately **18.38 units**. If you prefer the exact form, it remains \(\sqrt{338}\).
Answered by
GPT-4o mini
To compute the distance between points D and C using the Pythagorean theorem, we can recognize that the distance between two points in the Cartesian plane forms a right triangle with the differences in the x-coordinates and y-coordinates as the two legs.
Given:
- \(D(5, 11)\)
- \(C(10, -1)\)
We can calculate the differences:
1. Difference in x-coordinates:
\[
\Delta x = x_2 - x_1 = 10 - 5 = 5
\]
2. Difference in y-coordinates:
\[
\Delta y = y_2 - y_1 = -1 - 11 = -12
\]
Now, applying the Pythagorean theorem:
\[
d = \sqrt{(\Delta x)^2 + (\Delta y)^2}
\]
\[
d = \sqrt{(5)^2 + (-12)^2}
\]
\[
= \sqrt{25 + 144}
\]
\[
= \sqrt{169}
\]
\[
= 13
\]
Thus, the distance between points D and C is **13 units**.
Given:
- \(D(5, 11)\)
- \(C(10, -1)\)
We can calculate the differences:
1. Difference in x-coordinates:
\[
\Delta x = x_2 - x_1 = 10 - 5 = 5
\]
2. Difference in y-coordinates:
\[
\Delta y = y_2 - y_1 = -1 - 11 = -12
\]
Now, applying the Pythagorean theorem:
\[
d = \sqrt{(\Delta x)^2 + (\Delta y)^2}
\]
\[
d = \sqrt{(5)^2 + (-12)^2}
\]
\[
= \sqrt{25 + 144}
\]
\[
= \sqrt{169}
\]
\[
= 13
\]
Thus, the distance between points D and C is **13 units**.
Answered by
GPT-4o mini
The triangle symbols you are referring to (\( \Delta \)) are often used in mathematics to denote "change" or "difference" in a particular quantity. In the context of the distance formula and the Pythagorean theorem, \( \Delta \) represents the change in coordinates between two points:
- \( \Delta x \): This represents the change in the x-coordinates (the horizontal distance).
- \( \Delta y \): This represents the change in the y-coordinates (the vertical distance).
In the context of the distance calculation:
- \( \Delta x = x_2 - x_1 \) indicates how far apart the x-coordinates of the two points are.
- \( \Delta y = y_2 - y_1 \) indicates how far apart the y-coordinates of the two points are.
These differences are used to construct the legs of a right triangle, which ultimately allows us to use the Pythagorean theorem to find the distance between the two points (which is the hypotenuse of that triangle).
For reference:
- The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b):
\[
c^2 = a^2 + b^2
\]
In your case, the distance between points D and C can be modeled as a right triangle where \( \Delta x \) and \( \Delta y \) are the two legs, and the distance \( d \) is the hypotenuse.
- \( \Delta x \): This represents the change in the x-coordinates (the horizontal distance).
- \( \Delta y \): This represents the change in the y-coordinates (the vertical distance).
In the context of the distance calculation:
- \( \Delta x = x_2 - x_1 \) indicates how far apart the x-coordinates of the two points are.
- \( \Delta y = y_2 - y_1 \) indicates how far apart the y-coordinates of the two points are.
These differences are used to construct the legs of a right triangle, which ultimately allows us to use the Pythagorean theorem to find the distance between the two points (which is the hypotenuse of that triangle).
For reference:
- The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b):
\[
c^2 = a^2 + b^2
\]
In your case, the distance between points D and C can be modeled as a right triangle where \( \Delta x \) and \( \Delta y \) are the two legs, and the distance \( d \) is the hypotenuse.