# finding the area of abd

• December 11th 2011, 11:54 AM
slapmaxwell1
finding the area of abd
Attachment 23064

i understand how to find area and i understand that in an equilateral triangle all sides are equal. the trouble i am finding with this particular problem is that i do not know how to solve without using plug/chug. if there is a faster way of doing this problem i definitely would like to know...
• December 11th 2011, 12:49 PM
scounged
Re: finding the area of abd
Quote:

Originally Posted by slapmaxwell1
Attachment 23064
the trouble i am finding with this particular problem is that i do not know how to solve without using plug/chug.

What do you mean by that? Do you already know how to solve it?
• December 11th 2011, 12:51 PM
slapmaxwell1
Re: finding the area of abd
Quote:

Originally Posted by scounged
What do you mean by that? Do you already know how to solve it?

i thought i did, but its not working out...
• December 11th 2011, 01:07 PM
scounged
Re: finding the area of abd
Ok. In the problem it says that |AB| = 2|BC|. I'd solve this one by letting |BC| equal a, and calculate the area of the triangles ACE and ABD, in terms of a. Can you do that?
• December 18th 2011, 12:08 PM
slapmaxwell1
Re: finding the area of abd
ok im still having trouble with this problem...i figured out that one side is sqrt(5), but then thats it. so the sides of the shades triangle are 1*k, 2*k and sqrt(5)*k.

im using k because it is saying its a ratio. so what i am asking is this, how can i use this info to find the area of the triangle abd?
• December 18th 2011, 05:11 PM
Wilmer
Re: finding the area of abd
Ok; to start, you do not need triangle BCF; disregard it.

Let a = BC; then AB = 2a (ratio 1:2) ; still with me?
Let AC = b

Using triangle ACE:
since sides = b, area = b^2 SQRT(3) / 4 ; since that area is given as 20:
b^2 SQRT(3) / 4 = 20
b = SQRT[80 / SQRT(3)] ; that'll give you b = ~6.796

Using right triangle ABC:
b^2 = a^2 + (2a)^2
a = SQRT(b^2 / 5) ; that'll give you a = ~4.298

Using triangle ABD:
since sides = 2a, area = a^2 SQRT(3) ; that'll give you area = 16.00000000....

Hope you followed...come back with questions if not.
• December 19th 2011, 07:20 AM
Soroban
Re: finding the area of abd
Hello, slapmaxwell1!

Quote:

Code:

                  A            E                   o  *  *  *  o               *  |\        *             *    | \  20  *       D o      2|  \    *             *    |  \  *               *  | 1  \ *                 B o-----o C                   *  *                     * *                     o                     F
$ABC$ is a right triangle with $BC\!:\!AB \,=\,1\!:\!2$
$AC\!E, AB\!D, BC\!F$ are equilateral triangles.
The area of $\Delta AC\!E = 20.$
Find the area of $\Delta AB\1D.$

. . $(A)\;18\qquad (B)\;16 \qquad (C)\;15 \qquad (D)\;14 \qquad (E)\;12$

Pythagorus says: $AC = \sqrt{5}$

The ratio of the areas of two similar polygons
. . is the square of the ratios of their sides.

The ratio of the sides of $\Delta AB\!D$ and $\Delta AC\!E$ is. $2\!:\!\sqrt{5}$

The ratio of their areas is. $2^2\!:\!(\sqrt{5})^2 \:=\:4\!:\!5$

We have: . $\frac{\Delta AB\!D}{20} \:=\:\frac{4}{5} \quad\Rightarrow\quad \Delta AB\!D \:=\:16\;\hdots\;\text{answer (B)}$