Grades 9-10 Video Solutions 2025-2025_9-10

Video Transcription

Problem 18. The two shaded rectangles are congruent. Each of the shaded rectangles has an area of 4. What is the area of the large rectangle ABCD? So let's start by seeing  what we can conclude from the information we know. So first, observe that AFE and GFH are opposite angles, so they have the same value. Also, since we know that AEF and FAG  are both right angles, because of the rectangles, that means that these triangles AFE and GFH have all the same angles. And since these two rectangles are congruent, we also know  that AE and GH have the same length, and so AFE and GFH are congruent triangles. Now, we can see what we can get using this information. So our total goal is to find the area of the  rectangle ABCD. But we can break that up by first finding the area of the triangle ADC, which we know is half the total area of ABCD. So if we look at area of ADC and we try to  break it up into parts we know, we see that instead of looking at GFH here, we can replace GFH with the area of AFE, since those two triangles are congruent. So the area of ADC  is the same as the area of ADC minus the area of GFH plus the area of AFE. But now if we  look at this new area, this is just the rectangle ADGE and the triangle GHC. And we can easily find the area of both those pieces. We know from the problem that ADGE has area 4, and  GHC is just half of the rectangle GHCI. So it has area 4 divided by 2, which is 2. So now ADC  has area 6. And since we know that the area of ABCD is twice the area of ADC, it must have area 12. So the answer is D.

Video Summary

The problem involves finding the area of a large rectangle ABCD by using the given information about two congruent shaded rectangles, each with an area of 4. By analyzing congruent triangles formed within the setup, the solution establishes that the area of triangle ADC is 6. Since the large rectangle ABCD's area is twice that of triangle ADC, its total area is calculated to be 12. Thus, the area of rectangle ABCD is determined to be 12.