Grades 11-12 Video Solutions 2012-2012_11-12

Video Transcription

Square ABCD has sides of length 2. E and F are the midpoints of sides AB and AD respectively.  G is a point on CF such that 3CG equals 2GF. The area of triangle BEG is which of the following?  So here's the diagram. E is the midpoint of AB. F is the midpoint of AD. And the ratio of CG to GF is 2 to 3 which is what we have right here. So first let's copy over this  triangle up over here. So we get something like this. And now the area of this big triangle that we've made is the same as the area of our original square which is 4. So we want  to figure out the area of this triangle right here. In order to do that we can just find the fraction of the base of this big triangle and the fraction of the height of this big  triangle that we have. So we know that this base right here is 1 quarter the base of this triangle right here because that's just how we've constructed it. Now we just need to  find the ratio of this height to this height. So first we see that this ratio to this ratio, the ratio of this segment to this segment is 2 to 3. So let's just call them 2x and  3x for some value of x which we don't really care about. Since this triangle over here is just this one but copied over that tells us that this is 5x. And so the ratio of this  length to the total length is 8x to 10x which is 4 fifths. And that has to be the same as the ratio of this height to this height. So that means this height is 4 fifths of this  height. Okay so that means that our total area of this triangle is the area of this triangle times 1 quarter times 4 fifths which is 4 times 1 quarter times 4 fifths is 4 fifths.

Video Summary

The problem involves calculating the area of triangle BEG within square ABCD with side length 2, where E and F are midpoints of sides AB and AD, respectively. Point G lies on CF such that 3CG = 2GF. By analyzing the geometric relationships, we first determine the area of a larger triangle within the square, observing that the base of triangle BEG is one-quarter of this larger triangle's base, and its height is four-fifths of that triangle's height. Thus, the area of triangle BEG is the product of the full area (4) times these fractions, resulting in an area of \(\frac{4}{5}\).

Keywords

triangle area

square ABCD

midpoints

geometric relationships

triangle BEG