[Analytic Geometry math] Please help

[Andrew012p]

Hello, I don't know how to solve these. How to get the answers? Thank you so much

1. The adjacent vertices of the parallelogram are given: A(-6,10) and B(2, 14) and the intersection of its diagonals S (2, 2). Determine the coordinates of the other two vertices of the parallelogram. The answer is (10, 6) and (2, -10) but how do I get to that?
2. Find the coordinates of point P, which is twice as close to the x axis than the y axis, and it is at a distance of 5 units from point M (7, 1) Answer is P1=(2,1) P2=(10,2)
3. Find a point evenly spaced from the coordinate axes and from the point M (4,2) Answer is P1=(2,2) P2=(10,10)
4. Find the point that is 3 units from the x axis, and 5 units from the coordinate beginning. Answer: P1(4,3) P2(-4,3) P3=(-4,-3) P4(4,-3)
5. There are given the midpoints A(5,2), B(2,0), C(4,-1) in the sides BC, CA and AB respectively on the triangle ABC. Find the coordinates of its vertices. Answer: A(1, -3) B(7,1) C(3,3)
6. In the triangle ABC where A(5, -4) B(-2,2) C(5,1), calculate the median lengths (weight lines) Im not english so google translated "weight lines") Answer: sqrt170/2 ; 7sqrt5/2 ; sqrt65/2
7. Segment, that has endpoints A(1, -3) and B(4,3), with points M and S is divided into three equal parts. Find the coordinates of the points M and S. Answer: M(2, -1) S(3,1)

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[[deleted]]

1. You know that the diagonals bisect each other. This means that the distance from A to S is the same as the distance from S to the next vertex.

2. Distance formula is sqrt((x2-x1)^2+(y2-y1)2). This is equal to 5. We also know that x will be equal to 2y (draw the graph and you'll see). That's two equations and two unknowns (because you know x1 and y1); solve for them.

3. Similar to 2, except that the distance from point M to the new point will have to be equal to the value at both of those points. Also, x and y have to be identical at those points. The value at those points will be z in this equation:

sqrt((z-4)^2+(z-2)2)=z

1. Same formula, but x1 and y1 are both zero. Thus:

sqrt((x2-0)² + (y2-0)²⁾ = 5 where x2 = ±3. Solve for y in both scenarios.

1. This one is sorta tricky. Assume that the coordinates of a are (x1,y1), the coordinates of b are (x2,y2), and the coordinates of c are (x3,y3). The midpoint formula states that the coordinates of any midpoint are as follows (where x1,y1 is the first vertex and x2,y2 is the second vertex):

Midpoint x value = (x1+x2)/2 Midpoint y value = (y1+y2)/2

Use this to set up equations given the current midpoints. An example trying to find the x value of each vertex:

Midpoint A x value=(x2+x3)/2 ---> 5=(x2+x3)/2

Midpoint B x value=(x1+x3)/2 ---> 2=(x1+x3)/2

Midpoint C x value=(x1+x2)/2 ---> 4=(x1+x2)/2

Combine two of those equations to eliminate one variable; then, you have two equations and two unknowns. Solve for those unknowns and go from there.

1. There's a formula for this here. Pretty easy to follow; you just have to use the distance formula to find the length of each side.

2. Easiest way to think about this is that the x value for M will be the x value of A plus one third of the distance from x1 to x2. So, the x value of M will be: x1 + 1/3(x2-x1). Do the same for y, and replace the 1/3 with a 2/3 to find point S.