Video Transcript
If 𝐴𝐵 equals 30 centimeters, 𝐵𝐶 equals 40 centimeters, and 𝐴𝐶 equals 45 centimeters, find the ratio between the areas of the triangle 𝐴𝐸𝐷 and the triangle 𝐴𝐸𝐶.
Let’s begin by filling in the three length measurements that we’re given. 𝐴𝐵 is 30 centimeters, 𝐵𝐶 is 40 centimeters, and 𝐴𝐶 is 45 centimeters. In this problem, we need to find the ratio between the areas of these two triangles, 𝐴𝐸𝐷 and 𝐴𝐸𝐶. At the minute, we don’t actually have enough information to work out the areas, but let’s see how we would go about doing that. We can recall that the area of a triangle is calculated as half times the base 𝑏 times the height ℎ. If we created a perpendicular line from vertex 𝐴 to the line segment 𝐸𝐶 and defined it as ℎ, then this would be the perpendicular height of both triangles 𝐴𝐸𝐷 and 𝐴𝐸𝐶.
To work out the area of triangle 𝐴𝐸𝐷, we would multiply the base, which is the line segment 𝐸𝐷, by one-half and multiply it by the perpendicular height ℎ. For triangle 𝐴𝐸𝐶, the base length is 𝐸𝐶, and so the area of triangle 𝐴𝐸𝐶 is one-half times 𝐸𝐶 times ℎ. To write the areas as a ratio, we could write this as one-half 𝐸𝐷 times ℎ to one-half 𝐸𝐶 times ℎ. As we have one-half ℎ as a common factor in both of these areas, then this ratio can be written as 𝐸𝐷 to 𝐸𝐶. At the minute, we don’t have the length information to determine the ratio 𝐸𝐷 to 𝐸𝐶. So, let’s record this at the top of the page and clear some space to determine these line segments.
Let’s return to the given diagram. We can observe that the interior angle of triangle 𝐴𝐵𝐶 has been bisected because angle 𝐵𝐴𝐷 is congruent to angle 𝐷𝐴𝐶. We also have that an exterior angle of the same triangle has been bisected because angle 𝐹𝐴𝐸 is congruent to angle 𝐸𝐴𝐵. We can therefore use two of the angle bisector theorems, which gives us a ratio between the line segments relating to interior and exterior angle bisectors of a triangle. By the interior angle bisector theorem, we can write that 𝐷𝐶 over 𝐷𝐵 is equal to 𝐴𝐶 over 𝐴𝐵. And by the exterior angle bisector, we have that 𝐸𝐵 over 𝐸𝐶 is equal to 𝐴𝐵 over 𝐴𝐶.
Now, given that 𝐴𝐵 is 30 centimeters and 𝐴𝐶 is 45 centimeters, we can substitute in these lengths. This gives us that 𝐷𝐶 over 𝐷𝐵 equals 45 over 30, which simplifies to three over two. In the same way, 𝐸𝐵 over 𝐸𝐶 equals 30 over 45, which simplifies to two-thirds. We now have two equations. Let’s take a closer look at the second equation. If we look at the diagram, we know that the line segment 𝐸𝐶 is equal to the line segment of 𝐸𝐵 plus the line segment of 𝐵𝐶. We can substitute this into the second equation along with the fact that the length of 𝐵𝐶 is 40 centimeters to give us that 𝐸𝐵 over 𝐸𝐵 plus 40 is equal to two-thirds.
Now we have just one unknown in this equation, we can cross multiply to calculate 𝐸𝐵. This gives us three times 𝐸𝐵 is equal to two times 𝐸𝐵 plus 40, so three times 𝐸𝐵 is equal to two times 𝐸𝐵 plus 80. Then, subtracting two times 𝐸𝐵 from both sides gives us that 𝐸𝐵 is 80 centimeters. Now that we know this length of 80 centimeters, let’s return to the fact that we’re really trying to calculate the ratio of 𝐸𝐷 to 𝐸𝐶. We could observe that we have enough information to determine the length of 𝐸𝐶, but we still need to determine the length of 𝐸𝐷. And to do this, we really need to know the length of the line segment 𝐷𝐵.
Let’s see if we can now use equation one that we determined earlier. From the diagram, we can observe that 𝐷𝐵 plus 𝐷𝐶 is equal to 𝐵𝐶, so 𝐷𝐶 is equal to 𝐵𝐶 minus 𝐷𝐵. And as we were given that the length of 𝐵𝐶 is 40 centimeters, we can say that 𝐷𝐶 is equal to 40 minus 𝐷𝐵. We can then substitute this into equation one to give us 40 minus 𝐷𝐵 over 𝐷𝐵 is equal to three over two. By cross multiplying, we have two times 40 minus 𝐷𝐵 is equal to three times 𝐷𝐵. Simplifying then, we have 80 minus two times 𝐷𝐵 is equal to three times 𝐷𝐵. Then, by adding two 𝐷𝐵 to both sides, we get that 80 is equal to five times 𝐷𝐵. Finally then, we divide through by five such that 80 over five is equal to 𝐷𝐵. This means that 𝐷𝐵 must be 16 centimeters.
We now have enough information to work out the ratio 𝐸𝐷 to 𝐸𝐶. 𝐸𝐷 is equal to 80 centimeters plus 16 centimeters, which is 96 centimeters. The length of 𝐸𝐶 is equal to 80 centimeters plus 40 centimeters, which is 120 centimeters. And therefore being careful that we write the values in the correct order in the ratio, we can say that the ratio of 𝐸𝐷 to 𝐸𝐶 is 96 to 120. This simplifies to the ratio of four to five. And of course, even though we have calculated the ratio of 𝐸𝐷 to 𝐸𝐶, remember that we determined at the beginning of this problem that the ratio between the areas of the two triangles 𝐴𝐸𝐷 and 𝐴𝐸𝐶 is the same as that ratio. Therefore, the answer is the ratio four to five.
