Acc. Geom/Algebra Iiname Solutions

Acc. Geom/Algebra IIName Solutions .

Partitioning Line SegmentsPeriod_____Date______

Example #1: Point A is at-4on a number line, and point B is at8. Find a point, T,so that T

partitions the directed line segment A to B in a3:1ratio.

A B

-10 -8 -6 -4 -2 0 2 4 6 8 10

Solution: Note that the total distance from A to B is8 – (-4) =12. Since we need a3:1ratio,

divide this distance into4 equal lengths(12 ÷ 4 = 3). Going from point A,point T will

be located at the end of 3 lengths. Therefore, T = -4 + 3(3) = 5 .

1.) PointAis at1,and pointBis at10. Find a point,T,so thatTpartitions the

directed line segmentAtoBin a2:1ratio.

2.) PointAis at-2,and pointBis at13. Find a point,T,so thatTpartitions the

directed line segmentAtoBin a3:2ratio.

3.) PointAis at-4,and pointBis at8. Find a point,T,so thatTpartitions the

directed line segmentAtoBin a1:4ratio.

4.) PointAis at-9,and pointBis at9. Find a point,T,so thatTpartitions the

directed line segmentBtoAin a5:1ratio. (Did you notice how this problem is different?)

5.) PointAis at-8,and pointBis at6. Find a point,T,so thatTpartitions the

directed line segmentBtoAin a3:4ratio.

6.) PointAis at-3,and pointBis at9. Find a point,T,so thatTis25%of the way

fromAtoB. (Think: How does this question relate to ratios?)

7.) PointAis at-8,and pointBis at12. Find a point,T,so thatTis60%of the way

fromA toB.

8.) PointAis at2,and pointBis at7. Find a point,T,so thatTis30%of the way

fromBtoA.

Now going from one dimension (number line) to two dimensions (coordinate plane):

Example #2: Given the points A(-3,-6)and B(7,9). Find the coordinates of the point P on the

directed line segmentthat partitionsin the ratio2:3.

Solution: Apply the ratio techniques you used in one dimension on the number line problems to two

dimensions. Note that the total x-distance from A to B is7– (-3) = 10,so we need the

x-coordinate to be: (2/5)*10 + -3 = 1. Also, the total y-distance from A to B is

9– (-6) = 15,so we need the y-coordinate to be: (2/5)*15 + -6 = 0. Therefore, P= (1, 0).

9.) Given the points A(3,-4)and B(9,5). Find the coordinates of the point P on the

directed line segmentthat partitionsin the ratio2:1.

10.) Given the points A(-7,0)and B(9,8). Find the coordinates of the point P on the

directed line segmentthat partitionsin the ratio3:1.

11.) Given the points A(5,-2)and B(-2,12). Find the coordinates of the point P on the

directed line segmentthat partitionsin the ratio5:2. (What’s different?)

12.) Given the points A(-1,4)and B(6,-10). Find the coordinates of the point P on the

directed line segmentthat partitionsin a 3:4 ratio.

13.) Given the points A(-1,5)and B(6,0). Find the coordinates of the point P on the

directed line segmentthat is80%of the wayfromA toB.

(Think: How does this question relate to ratios?)

14.) Given the points A(2,-3)and B(-8,2). Find the coordinates of the point P on the

directed line segmentthat is33.33%of the wayfromB toA.

15.) Given the points A(5.4,-6)and B(-1.4,-2). Find the coordinates of the point P on

the directed line segmentthat partitionsin a 1:1 ratio. This is a midpoint.

But it is easier to average the x values and average the y values to find the midpoint!