Question

segment ab, shown below, is divided into 5 equal parts. point a is located at $(-6,7)$, and point d is located at $(-2,3)$. what are the coordinates of point f?
a. $(10,13)$
b. $(6,1)$
c. $(10,-1)$
d. $(2,-1)$

Explanation:

Step1: Find x-distance from A to D

$\Delta x = -2 - (-6) = 4$

Step2: Find y-distance from A to D

$\Delta y = 3 - 7 = -4$

Step3: Calculate per-segment x increment

From A to D is 3 segments, so $x_{\text{step}} = \frac{4}{3}$

Step4: Calculate per-segment y increment

$y_{\text{step}} = \frac{-4}{3}$

Step5: Find x-coordinate of F

A to F is 4 segments: $x_F = -6 + 4\times\frac{4}{3} = -6 + \frac{16}{3} = \frac{-18+16}{3}? No, correction: Wait, A to D is 3 intervals, so A to F is 4 intervals. Wait, recheck: A(-6,7) to D(-2,3): 3 steps. So each step x: $\frac{-2 - (-6)}{3} = \frac{4}{3}$, y: $\frac{3-7}{3} = \frac{-4}{3}$. Then F is 4 steps from A: $x = -6 + 4\times\frac{4}{3} = -6 + \frac{16}{3} = \frac{-18+16}{3}? No, error: Wait, A to D is 3 parts, so A to C is 1, C to D is 2, D to E 3, E to F 4, F to B 5. So D is 3 parts from A, F is 4 parts from A, so the ratio of AF/AD is 4/3. Use section formula: If F is 4/3 of the way from A to D? No, better: Find coordinates of B first, then F is 4/5 from A to B.

Alternative Step1: Let vector AB = 5(vector AD/3). Vector AD = (-2 - (-6), 3-7) = (4, -4). So vector AB = 5(4/3, -4/3) = (20/3, -20/3). Coordinates of B: A + AB = (-6 + 20/3, 7 - 20/3) = (2/3, 1/3)? No, that can't be. Wait no, A to D is 3 equal parts, so total parts 5, so AD is 3/5 of AB. So AB = (5/3)AD. AD vector is (4, -4), so AB vector is (5/34, 5/3(-4)) = (20/3, -20/3). Then B = (-6 + 20/3, 7 - 20/3) = (-18/3 +20/3, 21/3 -20/3) = (2/3, 1/3). No, that doesn't match options. Wait, mistake: A to D is 3 intervals, so from A to D is 3 steps, so each step is ( (x_D - x_A)/3, (y_D - y_A)/3 ) = ( (4)/3, (-4)/3 ). Then F is 4 steps from A: x = -6 + 4(4/3) = -6 + 16/3 = (-18 +16)/3 = -2/3? No, that's not an option. Wait, maybe count the segments wrong: A---C---D---E---F---B: that's 5 segments, so A to C is 1, C to D 2, D to E 3, E to F 4, F to B 5. So D is 2 segments from A, not 3! Oh right! That's the mistake. A to C is 1, C to D is 2, so D is 2 segments from A, F is 4 segments from A.

Step1: Correct segment count A to D

A to D is 2 equal segments.

Step2: Calculate x step per segment

$\Delta x_{\text{step}} = \frac{-2 - (-6)}{2} = \frac{4}{2} = 2$

Step3: Calculate y step per segment

$\Delta y_{\text{step}} = \frac{3 - 7}{2} = \frac{-4}{2} = -2$

Step4: Find x-coordinate of F

F is 4 segments from A: $x_F = -6 + 4\times2 = -6 + 8 = 2$

Step5: Find y-coordinate of F

$y_F = 7 + 4\times(-2) = 7 - 8 = -1$

Answer:

D. (2,-1)