Question

finding distance on the coordinate plane

1. plot the points
2. create a triangle
3. find the distance between the points

point a (-4, 1) point b (2, 4)
|-4 - 2| = 6 \t 6² + 3² = c²
|1 - 4| = 3 \t 36 + 9 = c²
\t\t\t√45 = 6.7
point c (-5, 2) point d (-2, -4)

point e (3,1) point f (2, -3)

point g (2,-2) point h (3, -4)

Explanation:

Step1: Recall distance formula

The distance $d$ between two points $(x_1,y_1)$ and $(x_2,y_2)$ is:
$$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$$

---

For Points A(-4,1) and B(2,4):

Step2: Substitute values into formula

$\sqrt{(2-(-4))^2+(4-1)^2}$

Step3: Simplify inside the root

$\sqrt{(6)^2+(3)^2}=\sqrt{36+9}=\sqrt{45}$

Step4: Simplify the radical

$\sqrt{45}=3\sqrt{5}\approx6.708$

---

For Points C(-5,2) and D(-2,-4):

Step2: Substitute values into formula

$\sqrt{(-2-(-5))^2+(-4-2)^2}$

Step3: Simplify inside the root

$\sqrt{(3)^2+(-6)^2}=\sqrt{9+36}=\sqrt{45}$

Step4: Simplify the radical

$\sqrt{45}=3\sqrt{5}\approx6.708$

---

For Points E(3,1) and F(2,-3):

Step2: Substitute values into formula

$\sqrt{(2-3)^2+(-3-1)^2}$

Step3: Simplify inside the root

$\sqrt{(-1)^2+(-4)^2}=\sqrt{1+16}=\sqrt{17}$

Step4: Calculate decimal approximation

$\sqrt{17}\approx4.123$

---

For Points G(2,-2) and H(3,-4):

Step2: Substitute values into formula

$\sqrt{(3-2)^2+(-4-(-2))^2}$

Step3: Simplify inside the root

$\sqrt{(1)^2+(-2)^2}=\sqrt{1+4}=\sqrt{5}$

Step4: Calculate decimal approximation

$\sqrt{5}\approx2.236$

Answer:

1. Distance between A(-4,1) and B(2,4): $3\sqrt{5}$ (or ≈6.708)
2. Distance between C(-5,2) and D(-2,-4): $3\sqrt{5}$ (or ≈6.708)
3. Distance between E(3,1) and F(2,-3): $\sqrt{17}$ (or ≈4.123)
4. Distance between G(2,-2) and H(3,-4): $\sqrt{5}$ (or ≈2.236)