Video: Using Properties of Reflection to Solve a Problem

If triangle 𝐴 is mapped by a reflection in the line 𝑦 = π‘₯ to triangle 𝐴′, would the two triangles be congruent?

04:23

Video Transcript

If triangle 𝐴 is mapped by a reflection in the line 𝑦 equals π‘₯ to triangle 𝐴 prime, would the two triangles be congruent?

Before we can answer this question, we need to define a few of these terms. Congruent means the same size and the same shape. And so, we’re asking if triangle 𝐴 and triangle 𝐴 prime are the same size and the same shape. We also need to consider what a reflection is. In a reflection, every point is the same distance from the central line. In our case, we have a reflection about the line 𝑦 equals π‘₯. And that means every point from triangle 𝐴 will be the same distance to the line 𝑦 equals π‘₯ as all the points in triangle 𝐴 prime. But let’s sketch this to see what it looks like.

First, we’ll sketch a coordinate plane. Now, let’s choose a triangle to be triangle 𝐴, with a point two, one; a point four, one; and a point three, two. And we’ll let this triangle be triangle 𝐴. Our next step will be to locate the reflection line 𝑦 equals π‘₯. We can use a table to find a few points on the line 𝑦 equals π‘₯. When π‘₯ equals one, since 𝑦 equals π‘₯, 𝑦 will be equal to one. And that means when π‘₯ is two, 𝑦 is two. And when π‘₯ is three, 𝑦 is three. Let’s go ahead and add those points to our line. One, one; two, two; three, three. We can sketch the line 𝑦 equals π‘₯ from these three points.

Our triangle 𝐴 prime needs to be reflected over this line. Something important happens to our coordinates when we reflect points across the line 𝑦 equals π‘₯. The π‘₯-coordinate and the 𝑦-coordinate change places. Since we have a point at two, one in our triangle 𝐴. We’ll have a point at one, two in the reflection. And since we have the point three, two in triangle 𝐴, we’ll have the point two, three in triangle 𝐴 prime. The final point in triangle 𝐴 is four, one which means we’ll have a point at one, four in triangle 𝐴 prime. So we connect the dots. And now we have to decide if these two triangles are congruent. Are they the same shape and the same size?

We know they’re the same shape. They’re both triangles. Let’s consider their base and their height. The reflection has a base from two to four and the height from one to two. The base from triangle 𝐴 goes from two to four and is also two units. The height of triangle 𝐴 is from one to two, is one unit. We know that the area of a triangle is one-half the height times the base. If we look at the areas of triangle 𝐴 and triangle 𝐴 prime, the area of triangle 𝐴 is one-half times two times one, one unit squared. And the area of triangle 𝐴 prime is one-half times two times one, one unit squared.

This confirms that these two triangles have the same shape and the same size. And so, we can say that, yes, these two triangles are congruent. In fact, it’s always true that a reflection has the same size as the original image. And so, we can say that a reflection is always congruent to its original image.

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