Question

what is the location of point f, which partitions the directed line segment from d to e into a 4:5 ratio?
-3
0
1
4

Explanation:

Step1: Identify coordinates of D and E

Point D is at \( x_D = -4 \), point E is at \( x_E = 5 \). The ratio is \( m:n = 4:5 \).

Step2: Use section formula

The formula for a point \( (x) \) dividing a line segment from \( x_1 \) to \( x_2 \) in ratio \( m:n \) is \( x = \frac{m \cdot x_2 + n \cdot x_1}{m + n} \).

Substitute \( x_1 = -4 \), \( x_2 = 5 \), \( m = 4 \), \( n = 5 \):

\( x = \frac{4 \cdot 5 + 5 \cdot (-4)}{4 + 5} \)

Step3: Calculate numerator and denominator

Numerator: \( 4 \cdot 5 + 5 \cdot (-4) = 20 - 20 = 0 \)

Denominator: \( 4 + 5 = 9 \)

So \( x = \frac{0}{9} = 0 \)? Wait, no, wait—wait, maybe I mixed up the ratio direction. Wait, the ratio is from D to E, so the formula is \( x = x_D + \frac{m}{m + n}(x_E - x_D) \).

Let's recalculate: \( x_E - x_D = 5 - (-4) = 9 \). Then the length of DE is 9 units. The ratio 4:5 means the first part (from D to F) is \( \frac{4}{4 + 5} \) of DE.

So \( x_F = x_D + \frac{4}{9} \times (x_E - x_D) = -4 + \frac{4}{9} \times 9 = -4 + 4 = 0 \)? Wait, but let's check again. Wait, \( x_D = -4 \), \( x_E = 5 \). The distance between D and E is \( 5 - (-4) = 9 \). The ratio 4:5 means the segment DF is 4 parts and FE is 5 parts, so total 9 parts, each part is 1 unit (since 9 parts = 9 units). So DF is 4 units. So starting at D (-4), moving 4 units towards E (which is in the positive direction, since E is at 5, which is to the right of D at -4). So -4 + 4 = 0. Wait, but let's use the section formula correctly. The section formula for internal division is \( x = \frac{m x_2 + n x_1}{m + n} \), where \( m:n \) is the ratio of DF:FE. So DF:FE = 4:5, so \( m = 4 \), \( n = 5 \), \( x_1 = -4 \) (D), \( x_2 = 5 \) (E). Then \( x = \frac{4 \times 5 + 5 \times (-4)}{4 + 5} = \frac{20 - 20}{9} = 0 \). Wait, but that gives 0. But let's check the answer options. 0 is an option. Wait, but let's verify with the number line. From D (-4) to E (5), the total length is 9. 4:5 ratio means 4 parts and 5 parts. Each part is 1 (since 9/9=1). So 4 parts from D: -4 + 41 = 0. Then 5 parts from F to E: 0 + 51 = 5, which is E. So that works. So the location of F is 0.

Answer:

0