Question

the points $(0,0)$, $(0,8)$, and $(5,0)$ are vertices of a right triangle. use the pythagorean theorem to calculate the exact distance between points $(0,8)$ and $(5,0)$. enter your answers in the boxes. the distance from $(0,8)$ to the origin is $square$ units. the distance from $(5,0)$ to the origin is $square$ units. use the pythagorean theorem. the distance from $(0,8)$ to $(5,0)$ is $square$ units.

Explanation:

Step1: Calculate distance from \((0,8)\) to origin \((0,0)\)

Use the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). Here \(x_1 = 0,y_1 = 0,x_2=0,y_2 = 8\).
\(d_1=\sqrt{(0 - 0)^2+(8 - 0)^2}=\sqrt{0 + 64}=8\)

Step2: Calculate distance from \((5,0)\) to origin \((0,0)\)

Use the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). Here \(x_1 = 0,y_1 = 0,x_2=5,y_2 = 0\).
\(d_2=\sqrt{(5 - 0)^2+(0 - 0)^2}=\sqrt{25+0}=5\)

Step3: Use Pythagorean theorem \(c=\sqrt{a^{2}+b^{2}}\)

Let \(a = 8\) (distance from \((0,8)\) to \((0,0)\)) and \(b = 5\) (distance from \((5,0)\) to \((0,0)\)). Then the distance \(c\) between \((0,8)\) and \((5,0)\) is \(c=\sqrt{8^{2}+5^{2}}=\sqrt{64 + 25}=\sqrt{89}\)

Answer:

\(8\)
\(5\)
\(\sqrt{89}\)