Question

what is the distance of the line segment between the endpoints (6, 8), (-7, 7)?
a. 13.5
b. 12
c. (-5, 7.5)
d. 13

Explanation:

Step1: Recall distance formula

The distance \(d\) between two points \((x_1,y_1)\) and \((x_2,y_2)\) is given by \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
Here, \((x_1,y_1)=(6,8)\) and \((x_2,y_2)=(-7,7)\).

Step2: Substitute values into formula

First, calculate \(x_2 - x_1=-7 - 6=-13\) and \(y_2 - y_1=7 - 8=-1\).
Then, substitute into the formula: \(d=\sqrt{(-13)^2+(-1)^2}\).

Step3: Simplify the expression

Calculate \((-13)^2 = 169\) and \((-1)^2 = 1\). So, \(d=\sqrt{169 + 1}=\sqrt{170}\)? Wait, no, wait, I made a mistake. Wait, \(y_2 - y_1 = 7 - 8=-1\), squared is 1. \(x_2 - x_1=-7 - 6=-13\), squared is 169. Wait, but 169 + 1 is 170? No, wait, no, the points are \((6,8)\) and \((-7,7)\). Wait, maybe I mixed up \(x_1\) and \(x_2\), but since squaring, it doesn't matter. Wait, but let's recalculate: \(x_2 - x_1=-7 - 6=-13\), \(y_2 - y_1=7 - 8=-1\). Then \((-13)^2=169\), \((-1)^2 = 1\), sum is 170? But 13 squared is 169, 13.04 squared is about 170. But the options are 13.5, 12, (-5,7.5), 13. Wait, maybe I made a mistake in the points. Wait, the second point is (-7,7)? Wait, maybe it's (-7, 7) and (6,8). Wait, let's check again. Wait, maybe the y-coordinate of the second point is 7? Wait, no, maybe I misread. Wait, the problem says endpoints (6,8) and (-7,7). Wait, but let's recalculate the distance formula correctly.

Wait, \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\)

\(x_1 = 6\), \(y_1 = 8\)

\(x_2=-7\), \(y_2 = 7\)

So \(x_2 - x_1=-7 - 6=-13\)

\(y_2 - y_1=7 - 8=-1\)

Then \((-13)^2 = 169\), \((-1)^2 = 1\)

Sum: \(169 + 1 = 170\)

\(\sqrt{170}\approx13.04\), which is approximately 13. So the answer is D.

Wait, maybe I made a mistake in the y-coordinate? Wait, maybe the second point is (-7, 7) or maybe (-7, -7)? Wait, no, the problem says (-7,7). Wait, but let's check again. Wait, if the second point was (-7, -7), then \(y_2 - y_1=-7 - 8=-15\), then \((-13)^2 + (-15)^2=169 + 225=394\), which is not. But the options have 13, which is close to \(\sqrt{170}\approx13.04\), so the answer is D.

Answer:

D. 13