SOLUTION: one of the side of a square lies along the straight line 4x+3y=26. the diagonals of the square intersect at the point(2,3).find the coordinates of the vertices,the equation of the

Question 975071: one of the side of a square lies along the straight line 4x+3y=26. the diagonals of the square intersect at the point(2,3).find the coordinates of the vertices,the equation of the sides of the square which are perpendicular to the given line,the equation of the circle passing through all the four vertices of the square.
Found 2 solutions by josgarithmetic, solver91311:
Answer by josgarithmetic(39620) (Show Source): You can put this solution on YOUR website!
Either draw, sketch, or visualize a 45-45-90 triangle that has two vertices on the given line and the other vertex being the given point (2,3). Use this to help find the distance from the given line to the given point (2,3). You have some special triangle knowledge to help you know the size of some identifiable segments of these triangles.

Al that is not a complete answer of any kind, but just a guide to how to think through this problem.

More Helpful Information As Guide:
What is the linear equation with slope and containing point (2,3) ? At what point does this line intersect with 4x+3y=26 ?
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!

First construct the given line, call that and the given point, call that (because it will be the center of the circle later in the problem).

Calculate the slope of and then using the idea that a perpendicular has a negative reciprocal slope, determine the slope of the line that passes through and is perpendicular to . Use the Point-Slope form of a line to write the equation of the perpendicular through , which we will call .

Solve the 2X2 system formed by the equations of and to find their point of intersection which we will call . Then use the distance formula to calculate the distance between and , a number we will call .

Using as a radius and as the center, write an equation for a circle centered at that passes through . This circle will intersect in two places, which we will call and , the order is arbitrary. Since the distance from to is half of the measure of our square and one side of the square lies in , segment represents one side of the circle.

Solve the 2X2 non-linear system consisting of the equation for and the equation for circle to determine the coordinates for and .

Using the distance formula, calculate the distance from to either or . (It doesn't matter which. Given correctly performed arithmetic, the answer will be the same either way and the arithmetic will be just as ugly). Call this distance .

Write the equation of a circle centered at with radius . This will be the circle that passes through all four vertices of the square.

Using the slope of line and point and the Point-Slope form of the equation of a line, derive the equation for the line containing the second side of the square, call this line . You already have one point of intersection between and circle ; use the two equations and solve the 2X2 non-linear system for the other point of intersection which we will call . Use a similar process to find the equation of the line parallel to through , and then solve the 2X2 system for the coordinates of .

The only thing remaining is to derive an equation for the line containing the fouth side of the triangle. Use the slope of and point or

John

My calculator said it, I believe it, that settles it

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