Question Video: Determining the Properties of the Diagonals of Squares | Nagwa Question Video: Determining the Properties of the Diagonals of Squares | Nagwa

Question Video: Determining the Properties of the Diagonals of Squares

What can you say about the diagonals of a square? [A] They are just perpendicular. [B] They are just equal in length. [C] They are perpendicular and equal in length. [D] They are neither perpendicular nor equal in length.

07:49

Video Transcript

What can you say about the diagonals of a square? Option (A) they are just perpendicular. Option (B) they are just equal in length. Option (C) they are perpendicular and equal in length. Or option (D) they are neither perpendicular nor equal in length.

Let’s begin by thinking about a square. The mathematical definition of a square is that it’s a quadrilateral or four-sided figure with all sides equal and all interior angles equal to 90 degrees. Therefore, if we were to draw a square, we know that in order to be a square, it’s got to have all four sides equal and all the angles inside are 90 degrees.

We could call this square 𝐴𝐵𝐶𝐷. Let’s consider the diagonals. Our first diagonal would be the line 𝐴𝐶 and the second diagonal is the line segment 𝐵𝐷. We might have a guess at some of the properties of these diagonals, but as we go through this question, we’ll need to prove all of our results. Let’s take a closer look at the first diagonal that we drew, the line segment 𝐴𝐶. We can consider it as part of this triangle 𝐴𝐷𝐶. So, what do we know about this triangle? Well, we know that the lengths 𝐴𝐷 and 𝐷𝐶 are equal in length and we know that there’s a right angle at angle 𝐴𝐷𝐶.

We can model the diagonal 𝐵𝐷 as part of a different triangle too. This time, we’ll have the triangle 𝐵𝐶𝐷. We know that 𝐵𝐶 and 𝐷𝐶 are equal in length as they’re part of the lengths of the square and the angle 𝐵𝐶𝐷 is 90 degrees. So, is there anything we can say about these two triangles that we’ve drawn?

We might wonder if these two triangles are in fact congruent, which means they’re the same shape and size. Don’t worry if you haven’t done too much on the congruency of triangles. But there is, in fact, a way in which we can prove that these triangles are congruent. Firstly, we have a pair of congruent sides. We have the side 𝐴𝐷 is equal in length to side 𝐵𝐶. There’s a second pair of corresponding sides. The line 𝐷𝐶 is a common side to both triangles, so that means we can say that they’re the same length.

Finally, we can say that there’s a pair of corresponding angles. The angle 𝐴𝐷𝐶 and the angle 𝐵𝐶𝐷 are both given as 90 degrees, so we know that they’re equal. We could, therefore, say that our two triangles 𝐴𝐷𝐶 and 𝐵𝐶𝐷 are congruent using the side-angle-side rule. As we have two congruent triangles, then the length 𝐴𝐶 in triangle 𝐴𝐷𝐶 will correspond with the length 𝐵𝐷 in triangle 𝐵𝐶𝐷. And so, we know that these two lengths will be equal. Remember that, of course, 𝐴𝐶 and 𝐵𝐷 were the diagonals of the square. So, we’ve just demonstrated that the diagonals of a square are equal in length.

There is another way we could have proved this result. That’s by using the Pythagorean theorem, which tells us that in right triangles, the square on the hypotenuse is equal to the sum of the squares on the other two sides. We can define the length 𝐴𝐷 with the letter 𝑥. And as 𝐷𝐶 is the same length — it was the side of the square — then this would also be 𝑥. In triangle 𝐵𝐶𝐷, the length 𝐷𝐶 is still 𝑥 units long and, therefore, so is the length 𝐵𝐶.

If we want to find the length of 𝐴𝐶, let’s say we call this 𝑦, and the length of 𝐵𝐷 which we call a different letter, let’s say we call it 𝑧, then we can apply the Pythagorean theorem. In triangle 𝐴𝐶𝐷, we would begin by saying that 𝑦 squared is equal to 𝑥 squared plus 𝑥 squared. We could simplify this to 𝑦 squared equals two 𝑥 squared. And then to find 𝑦, we would take the square root of both sides of this equation. So, 𝑦 would be equal to the square root of two 𝑥 squared.

Let’s have a look at triangle 𝐵𝐶𝐷. Here, we can write that 𝑧 squared is equal to 𝑥 squared plus 𝑥 squared. As before, we could work through solving this to find that 𝑧 is equal to the square root of two 𝑥 squared. What we’ve discovered here is that the diagonal 𝐴𝐶, which we defined as 𝑦, and the diagonal 𝐵𝐷, which we defined as 𝑧, are equal to the same value, once again showing that the diagonals of a square are equal in length.

The next thing we need to investigate about the diagonals of a square is whether or not they’re perpendicular. Remembering that the word “perpendicular” means at 90 degrees, that means we’re checking if all of these angles at the center would be 90 degrees. Let’s work through this investigation by taking two different triangles. If we label this point where the diagonals cross with the letter 𝐴, then this will help us with the labeling of our triangles.

Let’s consider the triangle on the left, 𝐴𝐸𝐷, and the triangle on the right, 𝐵𝐸𝐶. We remember that 𝐴𝐷 and 𝐵𝐶 are equal in length. But what else can we say about these two triangles? An important property of a square is that the opposite sides are parallel, so that will give us some clues about the angles in these triangles. If we look at this angle 𝐷𝐴𝐸, then it will be equal to the angle 𝐵𝐶𝐸. And this is because we have a pair of parallel lines and the transversal 𝐴𝐶, so these two angles will be alternate.

In the same way, angle 𝐴𝐷𝐸 will be equal to the angle 𝐶𝐵𝐸. We can also even say that angle 𝐴𝐸𝐷 is equal to angle 𝐵𝐸𝐶 as we have a pair of vertically opposite angles. We can, in fact, show that these two triangles are congruent. In order to prove congruency in triangles, we need to have at least one pair of sides congruent, which of course we have here. We know that 𝐴𝐷 will be equal in length to 𝐵𝐶 as they are two sides of the square. Using our two pairs of alternate angles and our pair of corresponding sides, we can then say that the two triangles are congruent by the angle-side-angle rule.

Let’s take a look at two other triangles in this square. We have triangle 𝐴𝐵𝐸 at the top and triangle 𝐷𝐸𝐶 underneath. Just like before, we have a pair of parallel and congruent sides. We’ll also have two pairs of alternate angles equal. Combined with the fact that we have a pair of corresponding sides equal, we could then say that these two triangles are congruent by the angle-side-angle rule. Let’s return to the original square and see how this information helps.

We have just shown that we have pairs of opposite triangles congruent. But remember that earlier we showed that if we split our square into a triangle along the diagonal, it will be congruent with the triangle created from the other diagonal. We could continue this pattern around the square to demonstrate that we must have four congruent triangles that meet at vertex 𝐸. The only way we could have four congruent triangles that all meet at vertex 𝐸 is if all of the angles created at 𝐸 are of 90 degrees. So, we can add to our note that the diagonals of a square are equal in length and perpendicular. This is a very useful property of a square, and it’s worthwhile remembering for examinations.

We can see that the answer given in option (C) is the correct one. The diagonals of a square are perpendicular and equal in length.

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