Question Video: Finding the Area of a Triangle Using the Relation between Triangles Sharing the Same Base and between Parallel Lines | Nagwa Question Video: Finding the Area of a Triangle Using the Relation between Triangles Sharing the Same Base and between Parallel Lines | Nagwa

Question Video: Finding the Area of a Triangle Using the Relation between Triangles Sharing the Same Base and between Parallel Lines Mathematics • Second Year of Preparatory School

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If the area of △𝐷𝐹𝐶 = 93.8 cm² and 𝐴𝐹 = 𝐹𝐸, find the area of △𝐴𝐵𝐸.

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Video Transcript

If the area of triangle 𝐷𝐹𝐶 equals 93.8 centimeters squared and 𝐴𝐹 equals 𝐹𝐸, find the area of triangle 𝐴𝐵𝐸.

First, let’s highlight triangle 𝐷𝐹𝐶. We know that the area of this space is 93.8 centimeters squared. We also know that the line 𝐹𝐴 and the line 𝐸𝐹 are equal in length. And we’re trying to find the area of triangle 𝐴𝐵𝐸, this space. However, it’s going to take a few steps for us to figure this out. So let’s go back to what we were originally given.

Before we can find the area of triangle 𝐴𝐵𝐸, let’s try to find other areas we can know inside this triangle. We know that if two triangles share the same base and their vertices lie on a straight line parallel to their base, then the two triangles are equal in area. If we consider the larger triangle, triangle 𝐴𝐷𝐶, and the triangle 𝐷𝐴𝐵, these two triangles have a shared base. And they have their vertices that lie on a straight line that is parallel to their base, which means the area of triangle 𝐴𝐷𝐶 is equal to the area of triangle 𝐷𝐴𝐵.

Now both of these triangles have the smaller triangle 𝐴𝐷𝐹 inside of them. If we take the area of triangle 𝐴𝐷𝐶 and subtract this smaller triangle, the area of triangle 𝐴𝐷𝐹, it would be equal to this space in orange, triangle 𝐷𝐹𝐶. And if we take away the area of triangle 𝐴𝐷𝐹, from the pink triangle 𝐷𝐴𝐵, it will be equal to the area in the pink, the area of triangle 𝐴𝐹𝐵. And we can say that these two areas are equal to each other. And since we know that the area of triangle 𝐷𝐹𝐶 equals 93.8 centimeters squared, we can say that the area of triangle 𝐴𝐹𝐵 is also equal to 93.8 centimeters squared.

Now that we’ve established that these two triangles have equal area, how does that help us? And what can that do so that we’ll be able to find the area of triangle 𝐴𝐵𝐸? We need another rule about triangles and their areas. We also know that if two triangles share the same vertex and have bases of equal length along the same line, then the heights of the triangles must be equal and the areas of the two triangles must be equal.

Remember that we’re looking for triangle 𝐴𝐵𝐸, this triangle. And inside triangle 𝐴𝐵𝐸, there are two smaller triangles, the triangle with its area highlighted in pink, triangle 𝐴𝐹𝐵, and the smaller triangle 𝐸𝐹𝐵. We noticed that both of these triangles share a vertex 𝐵. And they have bases of equal length along the same line, which tells us that the height of both of these triangles must be equal and their areas must also be equal.

The area of triangle 𝐴𝐹𝐵 equals the area of triangle 𝐸𝐹𝐵. We can say that the area we’re interested in, triangle 𝐴𝐵𝐸, must be equal to the area of triangle 𝐴𝐹𝐵 plus the area of triangle 𝐸𝐹𝐵, both of which equal 93.8 centimeters squared. 93.8 plus 93.8 equals 187.6 centimeters squared. Based on these two rules, we’ve shown that triangle 𝐴𝐵𝐸 has an area of 187.6 centimeters squared.

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