Question

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in the diagram, \\( \overline { a b } \\) is divided into equal parts. the coordinates of point \\( a \\) are \\( ( - 3,9 ) \\), and the coordinates of point \\( b \\) are \\( ( 9,5 ) \\).
the coordinates of point \\( c \\) are
the coordinates of point \\( e \\) are
the coordinates of point \\( h \\) are

Explanation:

Step1: Find the number of equal parts

From A to B, we can count the number of segments. Let's see: A to C is 1 segment, C to D is 1, D to E is 1, E to F is 1, F to G is 1, G to H is 1, H to I is 1, I to B is 1. Wait, actually, let's count the number of points. From A to B, how many intervals? Let's list the points: A, C, D, E, F, G, H, I, B. So there are 8 intervals (since between 9 points, there are 8 segments). Wait, no, let's check the coordinates. The total change in x: from A(-3,9) to B(9,5). The change in x is \(9 - (-3)=12\), change in y is \(5 - 9=-4\). If AB is divided into equal parts, let's see how many parts. Let's count the number of segments between A and B. From A to C: 1 segment, C to D: 2, D to E: 3, E to F:4, F to G:5, G to H:6, H to I:7, I to B:8. So 8 equal parts. So each part has a change in x of \(12\div8 = 1.5\) and change in y of \(-4\div8=-0.5\).

Step2: Coordinates of point C

Point C is 1 part away from A. So x-coordinate: \(-3+1.5\times1 = -1.5\), y-coordinate: \(9+(-0.5)\times1 = 8.5\). Wait, but maybe the number of parts is different. Wait, let's count the number of intervals between A and B. Let's see the points: A, C, D, E, F, G, H, I, B. So from A to B, there are 8 intervals (since 9 points, 8 gaps). So each interval is \(\frac{1}{8}\) of AB. Wait, but maybe the number of parts is 8? Wait, no, let's check the vector from A to B. The vector \(\overrightarrow{AB}=(9 - (-3),5 - 9)=(12,-4)\). If we divide AB into n equal parts, then each part is \(\frac{1}{n}\overrightarrow{AB}\). Let's see how many parts. Let's count the number of segments between A and C: 1, C to D:2, D to E:3, E to F:4, F to G:5, G to H:6, H to I:7, I to B:8. So n = 8. So each part is \(\frac{1}{8}\) of AB.

So for point C: it's 1 part from A. So the coordinates of C are \(A + \frac{1}{8}\overrightarrow{AB}\). So x: \(-3+\frac{12}{8}=-3 + 1.5=-1.5\), y: \(9+\frac{-4}{8}=9 - 0.5 = 8.5\). So C is (-1.5, 8.5) or (-3/2, 17/2).

Step3: Coordinates of point E

Point E is 3 parts from A (since A to C:1, C to D:2, D to E:3). So x: \(-3+\frac{12}{8}\times3=-3 + 4.5 = 1.5\), y: \(9+\frac{-4}{8}\times3=9 - 1.5 = 7.5\). So E is (1.5, 7.5) or (3/2, 15/2).

Step4: Coordinates of point H

Point H is 6 parts from A (A to C:1, C to D:2, D to E:3, E to F:4, F to G:5, G to H:6). So x: \(-3+\frac{12}{8}\times6=-3 + 9 = 6\), y: \(9+\frac{-4}{8}\times6=9 - 3 = 6\). So H is (6,6).

Wait, let's verify. Let's check the number of parts again. Wait, maybe I made a mistake in the number of parts. Let's count the number of segments between A and B. Let's list the points: A (1st), C (2nd), D (3rd), E (4th), F (5th), G (6th), H (7th), I (8th), B (9th). So the number of parts (intervals) between A and B is 8, so each part is \(\frac{1}{8}\) of AB. So the parameter t for point C is \(t = \frac{1}{8}\), D is \(t=\frac{2}{8}\), E is \(t=\frac{3}{8}\), F is \(t=\frac{4}{8}\), G is \(t=\frac{5}{8}\), H is \(t=\frac{6}{8}\), I is \(t=\frac{7}{8}\), B is \(t = 1\).

So for point C (t=1/8):

x: \(-3 + 12\times\frac{1}{8}=-3 + 1.5=-1.5\)

y: \(9 + (-4)\times\frac{1}{8}=9 - 0.5 = 8.5\)

Point E (t=3/8):

x: \(-3 + 12\times\frac{3}{8}=-3 + 4.5 = 1.5\)

y: \(9 + (-4)\times\frac{3}{8}=9 - 1.5 = 7.5\)

Point H (t=6/8=3/4):

x: \(-3 + 12\times\frac{3}{4}=-3 + 9 = 6\)

y: \(9 + (-4)\times\frac{3}{4}=9 - 3 = 6\)

So:

• Coordinates of C: (-1.5, 8.5) or (-3/2, 17/2)
• Coordinates of E: (1.5, 7.5) or (3/2, 15/2)
• Coordinates of H: (6, 6)

Answer:

• The coordinates of point C are \((-1.5, 8.5)\) (or \((-\frac{3}{2}, \frac{17}{2})\))
• The coordinates of point E are \((1.5, 7.5)\) (or \((\frac{3}{2}, \frac{15}{2})\))
• The coordinates of point H are \((6, 6)\)