Grades 9-10 Video Solutions 2023-2023_9-10

Video Transcription

Pentagon ABCDE is divided into four triangles with equal perimeter. Triangle ABC is equilateral and AEF, DFE, and CDF are three identical isosceles triangles.  What is the ratio of the perimeter of the pentagon ABCDE to the perimeter of triangle ABC? Let's call each side length of ABC having, let's say each side has length 2x.  Therefore we know that AF must have side length x. Then AEF and ABC we are given as the same side length, total side length, and therefore  because AF equals x we know that AE plus EF equals 5x and each of them are then 5 halves x. ABCDE's perimeter we can then find as DE plus AE plus CD plus AB plus BC.  We know DE will have side length x and AE plus DC will have side length 5x and then AB plus BC will be 4x.  So we have x plus 5x plus 4x giving a total of 10x as the perimeter of ABCDE and therefore the ratio that we want is 10x over 6x which is 5 thirds and our answer is D.

Video Summary

The problem involves finding the ratio of the perimeter of pentagon ABCDE to triangle ABC. Triangle ABC is equilateral with sides of length 2x. The remaining triangles are identical isosceles triangles, contributing to the total perimeter. Calculations determine that sides AE and CD are 5/2x each, while DE is x. Thus, the perimeter of ABCDE is x + 5x + 4x = 10x. The perimeter of triangle ABC is 6x. Therefore, the ratio of the pentagon's perimeter to triangle ABC's perimeter is 10x/6x, simplifying to 5/3. The answer is 5/3.

Keywords

perimeter ratio

pentagon ABCDE

triangle ABC

equilateral triangle

isosceles triangles