Point of Division

Point of Division

DIRECTION: Do the following as indicated. Use short bond papers for your solutions. To be submitted on Friday, Nov. 27, 2015.
1. Find the abscissa of the point on the x-axis that is units away from the point (2, 5).

2. Find the coordinates of the third vertex of an equilateral triangle that has vertices at (-1, 4) and (9, -6).

3. Find x so that the point (x, 5) is equidistant from (2, -1) and (-4, -2).

Suppose we want to find a point, say P, which is some fraction of the way from A to B. Let A = (x1, y1) and B = (x2, y2) be given and let P = (x, y) be the point we are seeking. If we let

Then P is 1/3 of the way from A to B when r = 1/3, P is 4/5 of the way from A to B when r = 4/5, and so on.
If A, B, and P are projected onto the x-axis to give the points Ax, Bx, and Px, respectively, we have, from elementary geometry,

Solving for x gives .
By projecting onto the y-axis, we have

1. Find the point one-third of the way from A(2, 5) to B(8, -1).
2. If the segment AB, where A(-3, 1) and B(2, 5), is extended beyond B to a point P four times as far from A as B is, find P.
3. If A = (-1, 5), B = (7, 1), and , find P.
4. The point (1, 4) is at a distance 5 units from the midpoint of the segment joining (3, -2) and ( x, 4). Find x.
5. The midpoints of the sides of a triangle are (-1, 3), (1, -2), and (5, -3). Find the vertices.

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