Determine whether BGHL is a rectangle given the set of vertices. B(0,5) G(4,7), H(5,4), L(1,2).

Question

Determine whether BGHL is a rectangle given the set of vertices. B(0,5) G(4,7), H(5,4), L(1,2).

Answer 1

Blake,

 

I think the easiest way to tackle these questions is to plot the vertices and look at the shape. You'll quickly see whether the given vertices form a rectangle - or something else entirely!

 

Each of those letters corresponds to a point on a graph. Do you know how to plot points? If you mark each of those vertices on a graph, you should see whether they form a rectangle. By the way, do you know what a vertex is? In this case, they correspond to a corner of the rectangle.

 

If you don't know how to plot points, the problem is a little more difficult. You'll have to use your knowledge of the properties of a rectangle, e.g., that its opposite sides are parallel, that each angle has 90 degrees, and so on. It's possible to determine whether BG and HL are parallel, or whether BL and GH are, just from the points. But it's a lot more work than looking at the graph.

 

Let me know if you need more hints.

 

Bryce

Answer 2

There's a couple of ways to do this, but basically you want to prove that the sides meet at right angles (and thus that opposite sides are parallel).  This can be done by finding the slope of each segment.

 

Remember that slope is change in y divided by change in x.  We want to show that BG and LH have the same slope, and that GH and BL have the same slope.  However the slopes for each pair need to be negative reciprocals of each other in order to show that they are perpendicular.  Then we have 2 sets of parallel sides that meet at right angles.

 

For example the slope of BG is (7-5)/(4-0) = 2/4 = 1/2

             and the slope of LH is (4-2)/(5-1) = 2/4 = 1/2

So they are parallel.

When you find the slopes of the other 2 segments, they should hopefully come out to -2 (the negative reciprocal of 1/2) proving the line segments are perpendicular to each other.  If they come out to be something different, then it is not a rectangle (but possibly a parallelogram).