Question

what is the distance between the points (-1, -3 3/4) and (-1, 4 1/2)

Answer 1

To find the distance between the two points \((-1, -3 \frac{3}{4})\) and \((-1, 4 \frac{1}{2})\), we first convert the mixed numbers to improper fractions or decimals for easier calculations:

- \(-3 \frac{3}{4} = -3 - \frac{3}{4} = -\frac{12}{4} - \frac{3}{4} = -\frac{15}{4}\)
- \(4 \frac{1}{2} = 4 + \frac{1}{2} = 4 + \frac{2}{4} = \frac{8}{2} + \frac{2}{4} = \frac{9}{2}\)

Thus, the points are \((-1, -\frac{15}{4})\) and \((-1, \frac{9}{2})\).

Since both points have the same x-coordinate of -1, we can calculate the distance using the y-coordinates only. The distance \(d\) between the two points is given by the absolute difference of the y-coordinates:

\[
d = \left| y_2 - y_1 \right|
\]

Substituting in the coordinates:

\[
d = \left| \frac{9}{2} - \left(-\frac{15}{4}\right) \right|
\]

To perform the subtraction, we first find a common denominator. The least common denominator of 2 and 4 is 4:

\[
\frac{9}{2} = \frac{18}{4}
\]

Now substitute:

\[
d = \left| \frac{18}{4} + \frac{15}{4} \right| = \left| \frac{18 + 15}{4} \right| = \left| \frac{33}{4} \right|
\]

Thus, the distance \(d\) is:

\[
d = \frac{33}{4}
\]

This can also be expressed as a mixed number:

\[
\frac{33}{4} = 8 \frac{1}{4}
\]

So, the distance between the points \((-1, -3 \frac{3}{4})\) and \((-1, 4 \frac{1}{2})\) is \(\frac{33}{4}\) or \(8 \frac{1}{4}\).