Do the diagonals of a parallelogram, or a rhombus, bisect each other?
Let’s start by reminding ourselves of what a parallelogram and what a rhombus look like. A parallelogram is a quadrilateral, a four-sided shape. It’s got two sets of parallel sides, and opposite sides are equal in length. We can mark on parallel sides on our diagram by using arrows corresponding to the sides that are parallel. And here we have a rhombus. A rhombus is a special type of parallelogram. So, it also has two sets of parallel sides opposite. But additionally, it has all four sides equal in length, which we can show in a diagram by a line through each length on the side.
In the question, we are asked if the diagonals bisect each other. The word bisect means to cut it exactly in half. For example, if we had a line that was 10 centimetres long, then bisecting the line would cut it into two pieces exactly five centimetres long. So, let’s look at the diagonals of our parallelogram and our rhombus. A diagonal is a line that goes from one corner to the corner opposite. As we’re being asked if the diagonals bisect each other, that means we need to check if both parts of the diagonal are the same length.
Let’s start by looking at the angles of our parallelogram. We could name off the vertices of our parallelogram as 𝐴, 𝐵, 𝐷, 𝐸, and the point where the diagonals cross as 𝐶. And this will help reference our angles as we go through. If we consider the angles at the center, we know that the measure of angle 𝐴𝐶𝐵 is equal to the measure of angle 𝐷𝐶𝐸. Because they’re vertically opposite angles. If we look at the angle 𝐵𝐴𝐸, we know that there will be another angle that’s equal to this one. And it is angle 𝐴𝐸𝐷. That’s because we have our parallel lines 𝐴𝐵 and 𝐷𝐸.
So, we have the two angles are equal because they’re alternate interior angles. And for the same reason, we know that the measure of angle 𝐴𝐵𝐷 will be equal to the measure of angle 𝐵𝐷𝐸. So, let’s take a closer look at the top triangle 𝐴𝐵𝐶 and the bottom triangle 𝐶𝐷𝐸. We’ve shown that they have three corresponding angles that are equal, but we also know that they have an equal side length. We know that the line 𝐴𝐵 in triangle 𝐴𝐵𝐶 is equal in length to the line 𝐷𝐸 in triangle 𝐶𝐷𝐸.
We know that because they’re part of a parallelogram, and so opposite sides are equal in length. Since we have three angles and a side, we can use the congruency rule angle angle side or A A S to show that these two triangles are congruent. So, now, we know we have two congruent triangles, triangle 𝐴𝐵𝐶 and triangle 𝐶𝐷𝐸. How will that help with the diagonals? Well, we already know that line 𝐴𝐵 is the same length as line 𝐷𝐸. Since the triangles are congruent, we know that line 𝐴𝐶 must be equal to line 𝐶𝐸. And that line 𝐵𝐶 is equal to line 𝐶𝐷. So, we can see that the diagonals of the parallelogram have cut each other exactly in half. That means that yes, they do bisect each other.
But let’s see if it’s any different with the rhombus. Well, as a rhombus also has all the properties of a parallelogram, we know that the angle properties will also be the same. However, in a parallelogram, we have two sets of congruent triangles. But in a rhombus, we have four congruent triangles. And this means that the angles created by the diagonals will be 90 degrees, so we can say that the diagonals in a rhombus are perpendicular by sectors. That is, they cut each other exactly enough at 90 degrees. And so, for our answer, we can say yes, the diagonals of a parallelogram or a rhombus do bisect each other.