Question

calculate the distance between the points (1,2) and (4,6).
a. 5 units
b. 4 units
c. 2 units
d. 3 units
which geometric figure is essential in explaining the distance formula using the pythagorean theorem?
a. triangle
b. ellipse
c. circle
d. parallelogram
what type of triangle is used to derive the distance formula from the pythagorean theorem?
a. right triangle
b. isosceles triangle
c. scalene triangle
d. equilateral triangle

Explanation:

Step1: Identify distance - formula variables

Let $(x_1,y_1)=(1,2)$ and $(x_2,y_2)=(4,6)$. The distance formula is $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.

Step2: Calculate differences

$x_2 - x_1=4 - 1=3$ and $y_2 - y_1=6 - 2 = 4$.

Step3: Apply the formula

$d=\sqrt{3^2 + 4^2}=\sqrt{9 + 16}=\sqrt{25}=5$.

For the second question:
The Pythagorean theorem $a^{2}+b^{2}=c^{2}$ is applied to a right - triangle. The distance formula between two points in a coordinate plane is derived from the Pythagorean theorem by considering the right - triangle formed by the two points and the horizontal and vertical displacements. So the geometric figure essential in explaining the distance formula using the Pythagorean theorem is a triangle.

For the third question:
The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ in a coordinate plane is derived from the Pythagorean theorem $a^{2}+b^{2}=c^{2}$, where the right - triangle is formed by the horizontal distance $(x_2 - x_1)$, the vertical distance $(y_2 - y_1)$ and the line segment connecting the two points. So the type of triangle used to derive the distance formula from the Pythagorean theorem is a right - triangle.

Answer:

1. A. 5 units
2. A. Triangle
3. A. Right triangle