### Video Transcript

If π΄π΅ equals 30 centimeters, π΅πΆ equals 40 centimeters, and π΄πΆ equals 45 centimeters, find the ratio between the areas of the triangle π΄πΈπ· and the triangle π΄πΈπΆ.

Letβs begin by filling in the three length measurements that weβre given. π΄π΅ is 30 centimeters, π΅πΆ is 40 centimeters, and π΄πΆ is 45 centimeters. In this problem, we need to find the ratio between the areas of these two triangles, π΄πΈπ· and π΄πΈπΆ. At the minute, we donβt actually have enough information to work out the areas, but letβs see how we would go about doing that. We can recall that the area of a triangle is calculated as half times the base π times the height β. If we created a perpendicular line from vertex π΄ to the line segment πΈπΆ and defined it as β, then this would be the perpendicular height of both triangles π΄πΈπ· and π΄πΈπΆ.

To work out the area of triangle π΄πΈπ·, we would multiply the base, which is the line segment πΈπ·, by one-half and multiply it by the perpendicular height β. For triangle π΄πΈπΆ, the base length is πΈπΆ, and so the area of triangle π΄πΈπΆ is one-half times πΈπΆ times β. To write the areas as a ratio, we could write this as one-half πΈπ· times β to one-half πΈπΆ times β. As we have one-half β as a common factor in both of these areas, then this ratio can be written as πΈπ· to πΈπΆ. At the minute, we donβt have the length information to determine the ratio πΈπ· to πΈπΆ. So, letβs record this at the top of the page and clear some space to determine these line segments.

Letβs return to the given diagram. We can observe that the interior angle of triangle π΄π΅πΆ has been bisected because angle π΅π΄π· is congruent to angle π·π΄πΆ. We also have that an exterior angle of the same triangle has been bisected because angle πΉπ΄πΈ is congruent to angle πΈπ΄π΅. 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 π·πΆ over π·π΅ is equal to π΄πΆ over π΄π΅. And by the exterior angle bisector, we have that πΈπ΅ over πΈπΆ is equal to π΄π΅ over π΄πΆ.

Now, given that π΄π΅ is 30 centimeters and π΄πΆ is 45 centimeters, we can substitute in these lengths. This gives us that π·πΆ over π·π΅ equals 45 over 30, which simplifies to three over two. In the same way, πΈπ΅ over πΈπΆ equals 30 over 45, which simplifies to two-thirds. We now have two equations. Letβs take a closer look at the second equation. If we look at the diagram, we know that the line segment πΈπΆ is equal to the line segment of πΈπ΅ plus the line segment of π΅πΆ. We can substitute this into the second equation along with the fact that the length of π΅πΆ is 40 centimeters to give us that πΈπ΅ over πΈπ΅ plus 40 is equal to two-thirds.

Now we have just one unknown in this equation, we can cross multiply to calculate πΈπ΅. This gives us three times πΈπ΅ is equal to two times πΈπ΅ plus 40, so three times πΈπ΅ is equal to two times πΈπ΅ plus 80. Then, subtracting two times πΈπ΅ from both sides gives us that πΈπ΅ is 80 centimeters. Now that we know this length of 80 centimeters, letβs return to the fact that weβre really trying to calculate the ratio of πΈπ· to πΈπΆ. We could observe that we have enough information to determine the length of πΈπΆ, but we still need to determine the length of πΈπ·. And to do this, we really need to know the length of the line segment π·π΅.

Letβs see if we can now use equation one that we determined earlier. From the diagram, we can observe that π·π΅ plus π·πΆ is equal to π΅πΆ, so π·πΆ is equal to π΅πΆ minus π·π΅. And as we were given that the length of π΅πΆ is 40 centimeters, we can say that π·πΆ is equal to 40 minus π·π΅. We can then substitute this into equation one to give us 40 minus π·π΅ over π·π΅ is equal to three over two. By cross multiplying, we have two times 40 minus π·π΅ is equal to three times π·π΅. Simplifying then, we have 80 minus two times π·π΅ is equal to three times π·π΅. Then, by adding two π·π΅ to both sides, we get that 80 is equal to five times π·π΅. Finally then, we divide through by five such that 80 over five is equal to π·π΅. This means that π·π΅ must be 16 centimeters.

We now have enough information to work out the ratio πΈπ· to πΈπΆ. πΈπ· is equal to 80 centimeters plus 16 centimeters, which is 96 centimeters. The length of πΈπΆ 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 πΈπ· to πΈπΆ is 96 to 120. This simplifies to the ratio of four to five. And of course, even though we have calculated the ratio of πΈπ· to πΈπΆ, remember that we determined at the beginning of this problem that the ratio between the areas of the two triangles π΄πΈπ· and π΄πΈπΆ is the same as that ratio. Therefore, the answer is the ratio four to five.