Question

Find the point that is 3/5 the distance from B to A B (-4,4)A (3,-4)

Answer 1

A point C between two other points A and B, in the line joining A to B, is a linear combination of A and B.

If C is a/n the distance from B to A, then it is (n-a)/n the distance from A to B. We can write that as:

`C=(a)/(n)A+(n-a)/(n)B`

Notice that the farther it is from B (as a increases), the larger is the factor (a/n) by which we multiply A.

Step 1

Identify the points A and B, and the fractions a/n and (n-a)/n.

We have:

`\begin{gathered} A=(3,-4) \\ B=(-4,4) \\ \\ (a)/(n)=(3)/(5) \\ \\ (n-a)/(n)=(5-3)/(5)=(2)/(5) \end{gathered}`

Step 2

Use the previous result in the formula to find C:

`\begin{gathered} C=(3)/(5)(3,-4)+(2)/(5)(-4,4) \\ \\ C=\mleft((9)/(5),-(12)/(5)\mright)+\mleft((-8)/(5),(8)/(5)\mright) \\ \\ C=\mleft((9-8)/(5),(-12+8)/(5)\mright) \\ \\ C=\mleft((1)/(5),-(4)/(5)\mright) \end{gathered}`

Answer

Therefore, the point that is 3/5 the distance from B to A is

`\mleft((1)/(5),-(4)/(5)\mright)`