Question

Segment AB in the coordinate plane has end points with coordinates A(5,4) and B(-10,-6). Graph line AB and find/plot two points C and D, so that they divide line AB into two parts with lengths in a ratio of 2:3. Also, provide the coordinates of the points of partition.

a) (0,0), (-5,-3)
b) (-5,-3), (0,0)
c) (2,1), (-5,-3)
d) (-5,-3), (2,1)

Answer 1

The correct answer is c) (2,1), (-5,-3).

Step-by-step explanation:

To find points C and D, we need to divide the line AB into two segments in a ratio of 2:3. The total ratio parts are 2 + 3 = 5. To determine the coordinates of C and D, we calculate the distance for each part of the ratio.

The x-coordinate of point C is found by taking 2/5 of the total x-distance from A to B and adding it to the x-coordinate of A:

`\[ x_C = 5 + (2)/(5) * (-10 - 5) = 5 + (2)/(5) * -15 = 2 \]`

Similarly, the y-coordinate of point C is found by taking 2/5 of the total y-distance from A to B and adding it to the y-coordinate of A:

`\[ y_C = 4 + (2)/(5) * (-6 - 4) = 4 + (2)/(5) * -10 = 1 \]`

So, the coordinates of point C are (2,1). The coordinates of point D can be found using the same process but with a ratio of 3/5. The calculations yield the coordinates of D as (-5,-3).

Therefore, the correct answer is option c) (2,1), (-5,-3).