Video Transcript
A light ray reflects from a mirror, as shown in the diagram. The length 𝐴𝐵 equals four centimeters, the length 𝐵𝐶 equals four centimeters, and the length 𝐶𝐸 equals five centimeters. What is the length 𝐷𝐸?
So, this question is about a light ray reflecting from a mirror. Looking at the diagram, this right here is the incident light ray and this here is the reflected ray. Finally, this gray rectangle here is the mirror that the light ray reflects from. We can recall that when light reflects from a surface, it does so according to a particular law. This law is known as the law of reflection, and it works as follows. At the point where the incident ray meets the reflecting surface, we can draw something called the normal to the surface.
In the diagram from the question, the normal to the surface is this dashed line. The normal is a line that is perpendicular to the surface. That is, it meets the surface at an angle of 90 degrees. The angle between the incident light ray and the normal to the surface is known as the angle of incidence and is commonly labeled as 𝜃 𝑖. Similarly, the angle between the reflected ray and the same normal to the surface is known as the angle of reflection and is often labeled as 𝜃 𝑟.
What the law of reflection says is that the angle of incidence is equal to the angle of reflection. Or in terms of symbols, the law says that 𝜃 𝑖 is equal to 𝜃 𝑟. If we look again at our diagram, we see that we can define two triangles. The first one here is drawn in orange and the second one in blue. Both of these triangles are right-angled triangles as they have one angle that is 90 degrees. We also know that the orange triangle has a second angle equal to 𝜃 𝑖 and the blue triangle has a second angle equal to 𝜃 𝑟.
Now, from our law of reflection, we know that 𝜃 𝑖 is equal to 𝜃 𝑟. So, since both of these angles have the same value, we’ll give them both the same label and we’ll call each of these angles 𝜃. We can also work out what the third angle must be for each of the triangles. To do this, we need to recall that the sum of internal angles in a triangle is equal to 180 degrees. So, for the orange triangle, we’ll label this third angle as 𝜙 one. We know that the sum of the internal angles for this triangle must be equal to 180 degrees. So, we have that 𝜃 plus 𝜙 one plus 90 degrees is equal to 180 degrees. Rearranging this expression, we get that 𝜙 one is equal to 180 degrees minus 90 degrees minus 𝜃, which we can also just write as 90 degrees minus 𝜃.
Now, we can do the same thing for the blue triangle. In this case, we’ll call the third angle 𝜙 two. Since the three internal angles of this blue triangle must add to 180 degrees, then we have that 𝜃 plus 𝜙 two plus 90 degrees is equal to 180 degrees. We can rearrange this and simplify it in exactly the same way as we did for the orange triangle. And we find that 𝜙 two is equal to 90 degrees minus 𝜃. So, 𝜙 one is equal to 90 degrees minus 𝜃, and 𝜙 two is also equal to 90 degrees minus 𝜃. This means that 𝜙 one must equal 𝜙 two.
So, for both of the triangles in our diagram, we’ll give this third angle the same label of 𝜙. It should now be clear that these two triangles are similar triangles, since each of the two triangles has the same three internal angles. For any two similar triangles, corresponding sides on those triangles are in the same ratio. Corresponding sides are sides that extend between the same two angles in each triangle. For example, in our diagram, the side 𝐴𝐵 extends between the angle marked 𝜙 and the angle of 90 degrees in the orange triangle. Meanwhile, the side 𝐷𝐸 extends between the angle marked 𝜙 and the angle of 90 degrees in the blue triangle.
Therefore, the sides 𝐴𝐵 and 𝐷𝐸 are corresponding sides. We can indicate that these two sides are corresponding sides by drawing a small single line through each side. Similarly, the side 𝐵𝐶 in the orange triangle corresponds to the side 𝐶𝐸 in the blue triangle. This is because each of these sides extends between an angle marked as 𝜃 and an angle of 90 degrees. Let’s indicate that these two sides are corresponding sides with two small lines on each side.
We’re told in the question that the length 𝐵𝐶 is equal to four centimeters and the length 𝐶𝐸 is equal to five centimeters. The ratio between these two sides is given by 𝐶𝐸 divided by 𝐵𝐶. And this ratio is five centimeters divided by four centimeters, which works out as a ratio of 1.2. Now because our two triangles are similar triangles, we know that corresponding pairs of sides must be in the same ratio. Since the sides 𝐴𝐵 and 𝐷𝐸 are also corresponding sides, we know that the ratio of 𝐷𝐸 divided by 𝐴𝐵 must also be equal to 1.2. Or taking this equation and multiplying both sides by the length 𝐴𝐵, we have that the length 𝐷𝐸 is equal to 1.2 times the length 𝐴𝐵.
We’re told in the question that the length 𝐴𝐵 is equal to four centimeters. And so, we can substitute that value in in place of 𝐴𝐵 in this equation. Doing this gives us that 𝐷𝐸 is equal to 1.2 times four centimeters. Evaluating this right-hand side gives a result of five centimeters.
And so, our answer to the question, “what is the length of 𝐷𝐸?” is that 𝐷𝐸 is equal to five centimeters.
