Univ. of Wisconsin
J.D. Univ. of Wisconsin Law school
Brian was a geometry teacher through the Teach for America program and started the geometry program at his school
The diagonals of a square are perpendicular bisectors of one another. As a result, Their intersection forms four right angles, and each diagonal is split into two congruent pieces. Therefore, if given the length of a diagonal, the length of one segment of that diagonal is half of the length of the entire diagonal.
Additionally, the diagonals of a square are angle bisectors. Therefore, the diagonals split the right angles of the square into two 45° angles.
In this problem we’re told that quadrilateral ABCD is a square. So we’ve got this quadrilateral and it’s a square. We’re also told that diagonal BD is equal to 14. We’re being asked to find angles X, Y and Z and the lengths of segment DE.
Well, let’s start off by finding angle Y, that’s going to be the easiest one. If I look at angle Y, I’m going to go over and see what I know about a square's diagonals. Well, it looks like the diagonals of a square intersect at a right angle. So they are perpendicular bisectors of each other.
So if we go back, I see that Y must be a right angle. So I’m going to write that Y is 90 degrees. If I look at X and Z, I’m going to guess that these two might be congruent but I don’t know what they’re going to equal. If I look at this diagonal something special happens with that diagonal to this angle B.
Well, let’s go over and take a look and see what we know. It looks like any diagonal on a square bisects that angle. So we know that if angle B is 90 degrees, X will be half and Z will be half and half of 90 degrees is 45. So I’m going to write that both of those angles must be 45 degrees.
The last step is to say, well if BD is equal to 14, the whole thing is equal to 14, we know that this is bisecting all of those diagonals. So if we’re trying to find DE, DE will be half of 14 or 7.
The key things here, diagonals of a square are perpendicular bisectors of each other and they bisect those vertex angles.