Grades 7-8 Video Solutions 2024-2024_7-8

Video Transcription

Problem number 13. A square has a side length of 10 meters. It is divided into parts by three straight line segments as shown.  The areas of the two shaded triangles are A and B. What is the value of A minus B?  So first, some things to remember from geometry is that the area of a triangle is 1 half base times height, and the area of a square is base times height.  Well, in a square, the base and the height are the same length, but in general this is true for all rectangles.  Now, let's make this picture larger and let's label an additional triangle C.  Instead of finding the areas of triangles A and B, let's find the area of this triangle in red, A and C together.  We'll notice that it is half the area of the entire square, and the reason is that it has the same base and the same height as the square,  and we know that the area of a triangle is 1 half the base and the height, and the square is just the base times the height.  So that means that A plus C together, this red triangle, is half the area of the entire square.  By the same logic, this blue triangle, which is B plus C, is also half the area, because it has the same base and it has the same height as the square.  And so that means that since A plus C and B plus C both take up half the area of the square,  then A plus C equals B plus C, because they're both equal to half the area of the square.  If we subtract C from both sides, we get A is equal to B, and if we subtract B from both sides, then we get A minus B equals to 0.  And since the question asked us, what is the value of A minus B, the answer is A, 0 meters squared.

Video Summary

The problem involves finding the difference in areas between two shaded triangles, A and B, within a square with a side length of 10 meters. By understanding that a triangle’s area is half the base times height, we can express larger triangle formations within the square. Both the red triangle (A + C) and the blue triangle (B + C) cover half of the square's area because they both share the square’s complete base and height. Therefore, A + C equals B + C. Subtracting common components shows A equals B, resulting in A - B equaling 0. Hence, the value of A minus B is 0 square meters.

Keywords

triangle area

square geometry

shaded triangles

area difference

mathematical proof