Finding the vertices of a square

Question

A square has centre (2/1) and one vertex is at (5/6)

Then there are three points given:

P(-3/4)

Q(-1,-4)

R(5/-4)

Q: Which of these points are vertices of the square?

The other vertices are Q and R, but the answer "Q and R" is not a choice for the solution

Choices are

P Q and R

P only

P and R only

Q only

P and Q only

 

Answer 1

Hi Liz, 

 

The correct answer is P and Q. See the following graph to get the idea.

 

https://www.desmos.com/calculator/a4owjrypmf

 

How to make the decision depends on the techniques you have available to you. Here is what I did on the graph above.

 

The vertices of the square must lie on the circle depicted, because the vertices of the square are equidistant from the center.

 

The diagonals of a square have the same length and are perpendicular to each other. If you write the equation of the line passing through (2,1) and (5,6) you have the equation of one of the diagonals. If you write the equation of a line passing through (2,1) with a slope that is the negative reciprocal of the first diagonal, you will have the equation of the 2nd diagonal.

 

To find vertices of the square you want to find out the points of intersection of the circle with each of those diagonals.

 

I have not put the equations showing in the above graph in slope-intercept form. You might want to do that and then use substitution into the equation of the circle to find points of intersection.