Grades 11-12 Video Solutions 2022-2022_11-12

Video Transcription

Two rectangles are inscribed inside a triangle ABC. The dimensions of the rectangles are 1x5 and 2x3 respectively as shown. What is the height of the triangle with base BC?  So we want to figure out the length of this red segment right here. So let's give some points some names because they're going to be useful.  So since this is a rectangle, we know that its two opposite sides are parallel and similarly for this taller rectangle here.  So that tells us that this triangle, this triangle, and the big triangle are all similar to each other.  So now that we have a bunch of similar triangles, we know that the ratio of the height to the base has to be the same across all three of the triangles.  So we know that AD to GH has to be equal to AE to IJ like this. And then we can just substitute in some of these known quantities. So like GH is equal to 3 and IJ is equal to 5.  And now AD is equal to AF minus this portion right here, which is 2. And AE is equal to AF minus this portion right here, which is 1.  So now we can just solve for AF because that's what we wanted all along. So cross multiply by the 3 and the 5 to get that 5 AF minus 10 is 3 AF minus 3.  Move the 3 AF over here, this 10 over here to get 7, and we get AF equals 7 over 2. So our final answer is B.

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