Video: Congruent Triangles in Rectangles

The diagonal of a rectangle divides its surface into two _ triangles. [A] different [B] congruent

03:18

Video Transcript

The diagonal of a rectangle divides its surface into two blank triangles. The options to fill in the blank space are different or congruent.

Two triangles are congruent if they are exactly the same shape and size. This means that the three angles in one triangle must be the same size as the three angles in the other. And the three side lengths in one triangle must be the same as the three side lengths in the other. So, what this question is asking us is, are the two triangles formed by the diagonal of a rectangle the same or different?

Well, here is a rectangle. We can then draw in one of the rectangle’s diagonals, connecting opposite corners together. The diagonal divides the rectangle up into two triangles, triangles one and two. And it’s these two triangles which we’re looking to determine whether or not they are the same or different. To answer this question, we need to recall a key property of rectangles. Which is that, opposite sides in a rectangle are of equal length.

So, the two shorter sides in this rectangle are the same length. And also, the other pair of opposite sides, the longer sides in this rectangle, are the same length. We can mark these sides in different colours on our diagram, orange for the longer sides and green for the shorter sides.

Now let’s have a look at what we know about these two triangles. Firstly, they have a common side length, the length of the orange side. They also have another common side length, the length of the green side. The pink side is common to both triangles. So, they also have a third common side length. This means that the three side lengths in triangle one are the same as the three side lengths in triangle two. And therefore, we have the SSS, or Side-Side-Side, congruency condition. So, we can conclude that these two triangles formed by the diagonal of a rectangle are congruent. And the reasoning we’ve used is Side-Side-Side.

We could also have proved this using other properties of our rectangle and other congruency conditions. For example, interior angles in a rectangle are always 90 degrees. Therefore, each of our triangles has a right angle within it, making them each right triangles. If we now look at just the green side and the orange side and the right angle, which is the included angle between these two sides, we could also argue that the two triangles are congruent using the SAS, Side-Angle-Side, congruency condition.

There is also a third way we could do this. Once we’ve identified that the two triangles are right triangles, we note that the pink side, the diagonal of the rectangle, is the hypotenuse of each triangle. We could then use one of the shorter sides in each triangle, in this case the side marked in orange, and the right angle, now marked as 𝑅, and the hypotenuse, marked as 𝐻, to use the RHS congruency condition, which stands for Right-angle-Hypotenuse-Side. This condition is specific to right-angled triangles. So, there are many ways that we can prove that the diagonal of a rectangle divides its surface into two congruent triangles.

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