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The United States Capitol is located at (2,−4)on a coordinate grid. The White House is located at (−10,16)on the same coordinate grid. Select the two points on the straight line between the United States Capitol and the White House such that the ratio is 1:3.

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The two different point on a segment joining the United States Capital and the White House such that the ratio of the shorter segments created by each is 1 : 3 are  C1 = (-1, 7) and C2 = (-7, 13).

What is the justification for the above?

At first, we need to compute the vector distance between A(x,y) = (2,4) and B(x,y) = (-10, 16) by following vectorial subtraction.

AB = B - A ............................1

Where  AB  is Vector Distance between A and B, Dimensionless.

A, B - Vector distance between each point and origin, dimensionless.

If we know that A(x, y) = (2, 4) and B(x,y) = (-10, 16), then we have the following results:

AB = (-10, 16) - (2, 4)

AB = (-10 -2, 16, -4)

AB = (-12, 12)

Note that we can find the location of an point inside the line segment by using the following vectorial equation:

C = A + r * AB......................2

Where

r -  Segment factor, dimensionless.

C- Location of resulting point, dimensionless.

There are two different options for the location of resulting point:  r1  = 1/4  and r2 = 3/4 Now we proceed to find each option:

r1 = 1/4

C1 = (2, 4) + 1/4 * (-12, 12)

C1 = (2,4) + (-3, 3)

C1 = (-1, 7)

R2 = 3/4

C2 = (2,4) + 3/4 * (-12,12)

= (-7, 13)

The two points on a stretch connecting the United States Capital and the White House where the ratio of the shorter segments formed by each is 1: 3 are C1 = (-1, 7) and (-7, 13)

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