Let ABCD be an isosceles trapezoid having parallel bases AB...

Let ABCD be an isosceles trapezoid having parallel bases AB and CD with AB > CD. Line segments from a point inside ABCD to the vertices divide the trapezoid into four triangles whose areas are 2, 3, 4, and 5 starting with the triangle with base CD and moving clockwise as shown in the diagram below. What is the ratio AB/CD ? D C 2 5 3 4 A B (A) 3 (B) 2 + ?2 (C) 1 + ?6 (D) 2?3 (E) 3?2

Transcript

00:01 Here you are given an isosceles trapezoid where your bases a, b, and d .c are parallel to each other.

00:10 And we're also told that a, b, the bottom base, is greater than c .d, which is the top base.

00:17 We also have this point within our trapezoid that creates four triangles inside the trapezoid with areas of two, three, four, and five.

00:29 That does tell me that the total area of the trapezoid is going to equal 14.

00:45 And just a side note, the trapezoid area formula is one -half times the sum of the bases times the height of the trapezoid.

01:01 We don't know any information right now about the bases or the height of the trapezoid, but we are able to use the areas of those triangles to help us out.

01:13 So i'm going to start with this top triangle with the area of two.

01:20 I'm going to call the distance from d to c base one.

01:30 So the area of a triangle is one half base times height.

01:34 So i know that two is equal to one half times the base.

01:44 Times some height, which i'm going to call h1.

01:48 And if i rearrange that formula, i'll get 4 over b1 is equal to h1.

01:57 And i'm going to do something similar with this bottom triangle that had an area of 4.

02:02 I'll call this base 2.

02:06 So 4 is equal to 1 1⁄2 times base 2.

02:12 Again, times some height we'll call that h2.

02:14 And rearrange that one to get 8 over b2 is equal to h2.

02:24 And so within my trapezoid, i know that h1 plus h2 is going to equal the total height of that trapezoid, which is helpful if i come back down here to my trapezoid formula...

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