Question

de has endpoints d(4, 11) and e(8, 9). point f divides de into two parts with lengths in a ratio of 3:1. what are the two possible locations of f? (7.5, 8) (7, 9.5) (5.5, 8) (5, 10.5) (3, 5) (6, 10) submit

Explanation:

To find the coordinates of a point that divides a line segment in a given ratio, we use the section formula. The section formula for a point \( F(x, y) \) that divides the line segment joining \( D(x_1, y_1) \) and \( E(x_2, y_2) \) in the ratio \( m:n \) is given by:

\[
x = \frac{mx_2 + nx_1}{m + n}, \quad y = \frac{my_2 + ny_1}{m + n}
\]

Here, \( D(4, 11) \) and \( E(8, 9) \), and the ratio is \( 3:1 \). We need to consider two cases: when \( F \) divides \( DE \) internally in the ratio \( 3:1 \) and when \( F \) divides \( DE \) externally in the ratio \( 3:1 \).

Step 1: Internal Division (Ratio \( 3:1 \))

For internal division, \( m = 3 \) and \( n = 1 \).

Calculate \( x \)-coordinate:

\[
x = \frac{3 \times 8 + 1 \times 4}{3 + 1} = \frac{24 + 4}{4} = \frac{28}{4} = 7
\]

Calculate \( y \)-coordinate:

\[
y = \frac{3 \times 9 + 1 \times 11}{3 + 1} = \frac{27 + 11}{4} = \frac{38}{4} = 9.5
\]

So, the coordinates of \( F \) for internal division are \( (7, 9.5) \).

Step 2: External Division (Ratio \( 3:1 \))

For external division, the formula is:

\[
x = \frac{mx_2 - nx_1}{m - n}, \quad y = \frac{my_2 - ny_1}{m - n}
\]

Here, \( m = 3 \) and \( n = 1 \).

Calculate \( x \)-coordinate:

\[
x = \frac{3 \times 8 - 1 \times 4}{3 - 1} = \frac{24 - 4}{2} = \frac{20}{2} = 10
\]
Wait, that doesn't match the options. Wait, maybe I made a mistake. Wait, the ratio could be \( 1:3 \) for the other direction. Let's check the ratio \( 1:3 \) for internal division.

Step 3: Internal Division (Ratio \( 1:3 \))

For ratio \( 1:3 \), \( m = 1 \) and \( n = 3 \).

Calculate \( x \)-coordinate:

\[
x = \frac{1 \times 8 + 3 \times 4}{1 + 3} = \frac{8 + 12}{4} = \frac{20}{4} = 5
\]

Calculate \( y \)-coordinate:

\[
y = \frac{1 \times 9 + 3 \times 11}{1 + 3} = \frac{9 + 33}{4} = \frac{42}{4} = 10.5
\]

So, the coordinates of \( F \) for ratio \( 1:3 \) are \( (5, 10.5) \).

Wait, let's verify the options. The options are \( (7.5, 8) \), \( (7, 9.5) \), \( (5.5, 8) \), \( (5, 10.5) \), \( (3, 5) \), \( (6, 10) \).

Wait, maybe I made a mistake in the ratio. Let's recheck.

Wait, the problem says "divides \( DE \) into two parts with lengths in a ratio of \( 3:1 \)". So, the two possible cases are:

1. \( DF:FE = 3:1 \) (internal division, ratio \( 3:1 \))
2. \( DF:FE = 1:3 \) (internal division, ratio \( 1:3 \)) or external division? Wait, no, if it's a line segment, it's internal division. Wait, maybe the ratio is along the segment, so two cases: \( F \) is closer to \( D \) or closer to \( E \).

Wait, let's recalculate for \( DF:FE = 3:1 \):

\( x = \frac{3 \times 8 + 1 \times 4}{3 + 1} = \frac{24 + 4}{4} = 7 \)

\( y = \frac{3 \times 9 + 1 \times 11}{3 + 1} = \frac{27 + 11}{4} = 9.5 \) → \( (7, 9.5) \)

For \( DF:FE = 1:3 \):

\( x = \frac{1 \times 8 + 3 \times 4}{1 + 3} = \frac{8 + 12}{4} = 5 \)

\( y = \frac{1 \times 9 + 3 \times 11}{1 + 3} = \frac{9 + 33}{4} = 10.5 \) → \( (5, 10.5) \)

So the two possible locations are \( (7, 9.5) \) and \( (5, 10.5) \).

Answer:

The two possible locations of \( F \) are \( (7, 9.5) \) and \( (5, 10.5) \).