Question Video: Finding the Ratio between the Areas of Two Triangles Using the Angle Bisector Theorem | Nagwa Question Video: Finding the Ratio between the Areas of Two Triangles Using the Angle Bisector Theorem | Nagwa

# Question Video: Finding the Ratio between the Areas of Two Triangles Using the Angle Bisector Theorem Mathematics • First Year of Secondary School

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If π΄π΅ = 30 cm, π΅πΆ = 40 cm, and π΄πΆ = 45 cm, find the ratio between the areas of the β³π΄πΈπ· and the β³π΄πΈπΆ.

07:21

### Video Transcript

If π΄π΅ equals 30 centimeters, π΅πΆ equals 40 centimeters, and π΄πΆ equals 45 centimeters, find the ratio between the areas of the triangle π΄πΈπ· and the triangle π΄πΈπΆ.

Letβs begin by filling in the three length measurements that weβre given. π΄π΅ is 30 centimeters, π΅πΆ is 40 centimeters, and π΄πΆ is 45 centimeters. In this problem, we need to find the ratio between the areas of these two triangles, π΄πΈπ· and π΄πΈπΆ. At the minute, we donβt actually have enough information to work out the areas, but letβ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 π times the height β. If we created a perpendicular line from vertex π΄ to the line segment πΈπΆ and defined it as β, then this would be the perpendicular height of both triangles π΄πΈπ· and π΄πΈπΆ.

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

Letβs return to the given diagram. We can observe that the interior angle of triangle π΄π΅πΆ has been bisected because angle π΅π΄π· is congruent to angle π·π΄πΆ. We also have that an exterior angle of the same triangle has been bisected because angle πΉπ΄πΈ is congruent to angle πΈπ΄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·πΆ over π·π΅ is equal to π΄πΆ over π΄π΅. And by the exterior angle bisector, we have that πΈπ΅ over πΈπΆ is equal to π΄π΅ over π΄πΆ.

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

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

Letβs see if we can now use equation one that we determined earlier. From the diagram, we can observe that π·π΅ plus π·πΆ is equal to π΅πΆ, so π·πΆ is equal to π΅πΆ minus π·π΅. And as we were given that the length of π΅πΆ is 40 centimeters, we can say that π·πΆ 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 πΈπ· to πΈπΆ. πΈπ· is equal to 80 centimeters plus 16 centimeters, which is 96 centimeters. The length of πΈπΆ 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 πΈπ· to πΈπΆ is 96 to 120. This simplifies to the ratio of four to five. And of course, even though we have calculated the ratio of πΈπ· to πΈπΆ, remember that we determined at the beginning of this problem that the ratio between the areas of the two triangles π΄πΈπ· and π΄πΈπΆ is the same as that ratio. Therefore, the answer is the ratio four to five.

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