Grades 11-12 Video Solutions 2023-2023_11-12

Video Transcription

Problem number 19. A pentagon is divided into smaller parts as shown. The numbers inside the triangles indicate their areas. What is the area P of the shaded quadrilateral?  Okay, so for this problem we're first going to add some variables so that it's easier to  refer to shapes and side lengths later on in the problem. We can see that instead of having numbers  we've replaced everything with variables for the area and some crucial side lengths have been  replaced by lowercase letters. Also, in the shaded region P we've drawn a line in the middle  making two different triangles X and Y. We can find the area of both X and Y and use that to  sum that and then find the area P. Notice that triangles A and B have the same height but different bases. Well, when you have two triangles with the same height and different bases  or two triangles with the same base but different heights their ratios of their area ratios are going  to just be the ratios of their differing base or differing height. So in this case the ratio  A to B is actually just equal to the lowercase ratio A over B because the height of the two  triangles is the same and so A over B is just 9 over 3 as we know from the original problem.  Triangles X and C also have the same height but they have bases A and B and so we already know  the ratio A over B so X over C is equal to A over B which is equal to 9 over 3. We already figured  this out and so X is equal to 3 times C. Lucky for us we know the value of C is equal to 2 so 3  times 2 gives us 6 as the area of X. Similarly triangles E and F have the same height but bases  C and D so E over F equals C over D equals 4 over 8. Same exact trick we just apply it again.  Now we use D and Y. We know that the ratio D over Y well since they have the same height we can just  find as the ratio of their bases so C over D and that's equal to 4 over 8. So Y is just equal to  2 times the value of D which we know is 5 so 2 times 5 equals 10. So X is equal to 6, Y is equal  to 10 and our answer is the sum of those two values which gives us 16. So the correct answer is C.

Video Summary

In this problem, a pentagon is divided into triangles, and the task is to find the area of a shaded quadrilateral, labeled as P, by using area relationships between triangles. By assigning variables to areas and side lengths, the ratios of those side lengths help to determine the areas. The problem uses the concept that triangles sharing a common height have area ratios equivalent to the ratios of their differing bases. Applying this, the areas of triangles X and Y are calculated as 6 and 10, respectively, leading to the sum, and thus the area of P is 16. The correct answer is C.

Keywords

pentagon

triangles

area

ratios

quadrilateral