Grades 7-8 Video Solutions 2025-2025_7-8

Video Transcription

Problem number 27. In the quadrilateral ABCD, the points N and K are marked on sides BC and AD  respectively, so that BN is equal to 2NC and AK equals KD. The area of triangle CKD is 2, and the area of triangle ABN is 6. What is the area of quadrilateral ABCD?  Before we start working on this problem, it's useful to know the following fact from geometry.  If two triangles have the same height, then the ratio of their areas is the ratio of their bases.  The reason for this is remembering the formula for the area of a triangle. The area is base  times height over 2, but if the height is the same, then the area is proportional to the length  of the base. Since the height is a constant, the only thing that can change is the base,  and the base will be proportional to the area. Now let's look at our diagram. I've reproduced on the right in a slightly larger scale, and I've drawn the additional diagonal AC.  Let's look at the first triangle ABC. We know that these two triangles are actually having  the same height because ABN and ANC have the same height, which is shown by the dotted line. Note that the height does not actually need to be in the triangle. In this case,  since the triangle is obtuse, the height lies outside the triangle, but it doesn't matter  because the area is the base times height over 2, regardless of if the height is inside or outside  the triangle. Now remember from the prompt statement that BN is equal to 2 times NC.  That is, BN is twice NC. Now, since the bases are in the 2 to 1 ratio, the areas must also be in a  2 to 1 ratio. Since the area of ABN is 6, the area of ANC has to be half of that, or 3. We can do  the same logic with triangles AKC and CKD. We know that these two bases are the same, since AK  equals KD from the prompt statement, and they both share the same height. Again, note that this height  is outside triangle CKD, but it doesn't matter for the problem because it just says that the area  is proportional to the ratio of the bases. In this case, both the base and the height are the same,  so in fact the areas of these two triangles should be the exact same. That is, the area of  triangle AKC should be equal to the area of triangle CKD, which is 2. Now, we know the area  of all four regions of the quadrilateral ABCD, so we can add them up. 6 plus 3 plus 2 plus 2, and if we do that, we get 13. That means that the area of quadrilateral ABCD is A, 13.

Video Summary

Summary Not Available