When you use the distance formula, you are building a geometric figure whose hypotenuse connects two given points. Let's explore this step-by-step:
1. The distance formula is derived from a geometric property related to a specific shape. The formula is given by:
[tex]\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
where [tex]\( (x_1, y_1) \)[/tex] and [tex]\( (x_2, y_2) \)[/tex] are the coordinates of the two points.
2. The formula directly uses the coordinates to calculate the straight-line distance between the two points.
3. To derive this formula, consider two points, [tex]\( (x_1, y_1) \)[/tex] and [tex]\( (x_2, y_2) \)[/tex], plotted on a Cartesian plane.
4. Draw a horizontal line from [tex]\( (x_1, y_1) \)[/tex] to [tex]\( (x_2, y_1) \)[/tex], and then a vertical line from [tex]\( (x_2, y_1) \)[/tex] to [tex]\( (x_2, y_2) \)[/tex]. This forms a right triangle where:
- The horizontal segment measures [tex]\( |x_2 - x_1| \)[/tex]
- The vertical segment measures [tex]\( |y_2 - y_1| \)[/tex]
5. The hypotenuse of this right triangle is the straight-line distance between [tex]\( (x_1, y_1) \)[/tex] and [tex]\( (x_2, y_2) \)[/tex].
6. According to the Pythagorean theorem, for this right triangle:
[tex]\[
\text{Hypotenuse}^2 = (\text{Base})^2 + (\text{Height})^2
\][/tex]
which translates to:
[tex]\[
d^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2
\][/tex]
Taking the square root of both sides gives us:
[tex]\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
Therefore, when you use the distance formula, you are essentially working with a right triangle whose hypotenuse is the distance between the two given points.
The correct answer is:
D. right triangle
