Question Video: Identifying the Triangle with the Same Area as a Given Triangle Using the Relations between Triangles Sharing the Same Base and between Parallel Lines Mathematics • Second Year of Preparatory School

Video Transcript

Which triangle has the same area as triangle 𝐵𝐿𝐴?

The area of triangle 𝐵𝐿𝐴 is the combined areas of triangles 𝐵𝐿𝐺 and 𝐵𝐺𝐴. Notice that the segment 𝐵𝐺 is a median of the triangle. This means that triangles 𝐵𝐿𝐺 and 𝐵𝐺𝐴 have the same area. This is because their bases are congruent and subsets of the same line, and they share a common vertex at 𝐵, giving them bases and perpendicular heights of equal magnitude.

Triangles 𝐵𝐶𝐺 and 𝐾𝐶𝐺 have a common base, 𝐶𝐺. Furthermore, their vertices both lie on the line 𝐾𝐵, which is parallel to their base. This means that 𝐵𝐶𝐺 and 𝐾𝐶𝐺 have the same area. If we subtract the area of triangle 𝐿𝐶𝐺 from this common area, we find that triangle 𝐾𝐶𝐿 has the same area as 𝐵𝐿𝐺 and 𝐵𝐺𝐴.

Finally, observe that segment 𝐾𝐶 is a median of triangle 𝐾𝑂𝐿. This means that 𝐾𝑂𝐶 has the same area as 𝐾𝐶𝐿, 𝐵𝐿𝐺, and 𝐵𝐺𝐴. We conclude that triangle 𝐾𝑂𝐿 has the same area as triangle 𝐵𝐿𝐴.