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 𝐴𝐡 = 30 cm, 𝐡𝐢 = 40 cm, and 𝐴𝐢 = 45 cm, find the ratio between the areas of the △𝐴𝐸𝐷 and the △𝐴𝐸𝐢.

07:21

Video Transcript

If 𝐴𝐡 equals 30 centimeters, 𝐡𝐢 equals 40 centimeters, and 𝐴𝐢 equals 45 centimeters, find the ratio between the areas of the triangle 𝐴𝐸𝐷 and the triangle 𝐴𝐸𝐢.

Let’s begin by filling in the three length measurements that we’re given. 𝐴𝐡 is 30 centimeters, 𝐡𝐢 is 40 centimeters, and 𝐴𝐢 is 45 centimeters. In this problem, we need to find the ratio between the areas of these two triangles, 𝐴𝐸𝐷 and 𝐴𝐸𝐢. 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 β„Ž. If we created a perpendicular line from vertex 𝐴 to the line segment 𝐸𝐢 and defined it as β„Ž, then this would be the perpendicular height of both triangles 𝐴𝐸𝐷 and 𝐴𝐸𝐢.

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 β„Ž. For triangle 𝐴𝐸𝐢, the base length is 𝐸𝐢, and so the area of triangle 𝐴𝐸𝐢 is one-half times 𝐸𝐢 times β„Ž. To write the areas as a ratio, we could write this as one-half 𝐸𝐷 times β„Ž to one-half 𝐸𝐢 times β„Ž. As we have one-half β„Ž as a common factor in both of these areas, then this ratio can be written as 𝐸𝐷 to 𝐸𝐢. At the minute, we don’t have the length information to determine the ratio 𝐸𝐷 to 𝐸𝐢. 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 𝐴𝐡𝐢 has been bisected because angle 𝐡𝐴𝐷 is congruent to angle 𝐷𝐴𝐢. We also have that an exterior angle of the same triangle has been bisected because angle 𝐹𝐴𝐸 is congruent to angle 𝐸𝐴𝐡. 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 𝐷𝐢 over 𝐷𝐡 is equal to 𝐴𝐢 over 𝐴𝐡. And by the exterior angle bisector, we have that 𝐸𝐡 over 𝐸𝐢 is equal to 𝐴𝐡 over 𝐴𝐢.

Now, given that 𝐴𝐡 is 30 centimeters and 𝐴𝐢 is 45 centimeters, we can substitute in these lengths. This gives us that 𝐷𝐢 over 𝐷𝐡 equals 45 over 30, which simplifies to three over two. In the same way, 𝐸𝐡 over 𝐸𝐢 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 𝐸𝐢 is equal to the line segment of 𝐸𝐡 plus the line segment of 𝐡𝐢. We can substitute this into the second equation along with the fact that the length of 𝐡𝐢 is 40 centimeters to give us that 𝐸𝐡 over 𝐸𝐡 plus 40 is equal to two-thirds.

Now we have just one unknown in this equation, we can cross multiply to calculate 𝐸𝐡. This gives us three times 𝐸𝐡 is equal to two times 𝐸𝐡 plus 40, so three times 𝐸𝐡 is equal to two times 𝐸𝐡 plus 80. Then, subtracting two times 𝐸𝐡 from both sides gives us that 𝐸𝐡 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 𝐸𝐢. We could observe that we have enough information to determine the length of 𝐸𝐢, 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 𝐷𝐡.

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

We now have enough information to work out the ratio 𝐸𝐷 to 𝐸𝐢. 𝐸𝐷 is equal to 80 centimeters plus 16 centimeters, which is 96 centimeters. The length of 𝐸𝐢 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 𝐸𝐢 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 𝐸𝐢, remember that we determined at the beginning of this problem that the ratio between the areas of the two triangles 𝐴𝐸𝐷 and 𝐴𝐸𝐢 is the same as that ratio. Therefore, the answer is the ratio four to five.

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