Question

Find the coordinates of the point 13
of the way from A to B.

The graphs with X-coordinate marks -4, 0, 4, 8 and 12. Y-coordinate marks -8, -4, 0, and 4. There is a line segment AB passes through points A (-3,-2) and B (12, 4).


coordinates = (
,
)

Answer 1

To find the coordinates of the point that is \( \frac{13}{20} \) of the way from point A to point B, we can use the formula for finding a point dividing a segment in a specific ratio.

The coordinates of points A and B are:
\[ A(-3, -2) \]
\[ B(12, 4) \]

Let \( P(x, y) \) be the point \( \frac{13}{20} \) of the way from A to B.

Using the section formula, where \( P \) divides the segment \( AB \) in the ratio \( m:n \), we can write the coordinates of \( P \) as follows:

\[
x = \frac{n \cdot x_A + m \cdot x_B}{m+n}
\]
\[
y = \frac{n \cdot y_A + m \cdot y_B}{m+n}
\]

In this case, moving from A to B, the ratio can be set as \( 13:7 \) because \( \frac{13}{20} \) of the distance means that the remaining part is \( 20 - 13 = 7 \).

Thus, we have:

- \( m = 13 \)
- \( n = 7 \)

Now substituting the coordinates of A and B:

### For \( x \):
\[
x = \frac{7 \cdot (-3) + 13 \cdot 12}{13 + 7}
\]
\[
= \frac{-21 + 156}{20}
\]
\[
= \frac{135}{20} = 6.75
\]

### For \( y \):
\[
y = \frac{7 \cdot (-2) + 13 \cdot 4}{13 + 7}
\]
\[
= \frac{-14 + 52}{20}
\]
\[
= \frac{38}{20} = 1.9
\]

Thus, the coordinates of the point that is \( \frac{13}{20} \) of the way from A to B are:

\[
\text{coordinates} = (6.75, 1.9)
\]