Question

the distance between the points (10, y) and (4, 8) is √61. arrange the equations below to show the process for finding the value of y.
√61 = √(10 - 4)²+(y - 8)²
√61 = √36+(y - 8)²
61 = 36+(y - 8)²
25=(y - 8)²
±5=y - 8
y = 13 or y = 3

Explanation:

Step1: Apply distance formula

The distance $d$ between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d = \sqrt{(x_1 - x_2)^2+(y_1 - y_2)^2}$. Here, $x_1 = 10$, $y_1=y$, $x_2 = 4$, $y_2 = 8$ and $d=\sqrt{61}$. So $\sqrt{61}=\sqrt{(10 - 4)^2+(y - 8)^2}$.

Step2: Simplify the equation

First, calculate $(10 - 4)^2=36$. The equation becomes $\sqrt{61}=\sqrt{36+(y - 8)^2}$. Square both sides to get $61=36+(y - 8)^2$.

Step3: Isolate the squared - term

Subtract 36 from both sides: $61-36=(y - 8)^2$, so $25=(y - 8)^2$.

Step4: Solve for y

Take the square root of both sides: $y - 8=\pm5$. When $y - 8 = 5$, $y=13$; when $y - 8=-5$, $y = 3$.

Answer:

$y = 13$ or $y = 3$