Grades 9-10 Video Solutions 2021-video 2021 9-10/25

Video Transcription

Problem number 25 states, the smallest square in the picture is an area of 16 and the gray triangle has an area of 1. What is the area of the larger square?  So first let's label all of our vertices. We'll label them A through G. So let's review what the problem told us. They tell us that square ABCD's area is equal to 16.  And that triangle GDF's area is equal to 1. And we want to know what the area of square GAFE is. First let's take a look at triangle AED.  The area of this triangle is 1 half base times height. And we know what the base times height is in this case because the base and height would be the side lengths of the square.  The triangle is in and we know the area of the square so we know the base times the height. In this case it's equal to 16. So the area of AED is equal to 1 half 16 which is 8.  Now let's consider triangle GDF. Let's look at the base and the height of the two triangles.  We can see that the sum of their altitudes is equal to the height of the square and their widths are also equal to the width of the square.  So this means the sum of their areas is going to be equal to half the area of the square. So algebraically AED plus GDF is equal to half of GAFE. So 8 plus 1 is half of GAFE.  Dividing both sides by 2 we get that GAFE is equal to 18. So the question asked us, what is the area of the larger square? The answer is 18. Letter B.

Video Summary

The problem involves two squares and a triangle. The smallest square, ABCD, has an area of 16, while triangle GDF has an area of 1. The task is to find the area of the larger square, GAFE. By calculating the area of triangle AED using the base and height derived from the smaller square's measurements, we find it equals 8. Noticing the relationship between the triangles and the larger square, it is established that 8 plus 1 equals half of the larger square's area. Solving for the larger square's area gives 18. Hence, the area of the larger square is 18.

Keywords

geometry

area calculation

squares and triangle

mathematical problem

larger square area