Question

The figure below is made up of Rectangle ABCD and Triangle AED. Shaded part x is 12 square centimetres smaller than shaded part Y. What is the length of AB?

Answer 1

1. Check the picture attached.

2. Triangles OAB and OEC are similar as they have congruent angles, as shown in the figure.

3. Similarity means that there is a scale factor of similarity, say k.

4.

`(12-t)/(t)=k` (corresponding sides, each in front of angles alpha)

also


`(9-a)/(a)=k` (corresponding sides in front of the vertec angles AOB and EOC)

so
`(12-t)/(t)=(9-a)/(a)`


`(12)/(t)-1= (9)/(a)-1`


`(12)/(t)= (9)/(a)`


`(4)/(t)= (3)/(a)`


`t= (4a)/(3)`

5.
x is y-12, so let's find the areas x and y, in terms of a, so that we form an equation and solve for a.

x =
`(1)/(2)at= (1)/(2)a* (4a)/(3)= ( 2a^(2) )/(3)`

To find the area y we need to write OC in terms of a:

OC=
`t*k=(4a)/(3)* (9-a)/(a)= (36-4a)/(3)`

so y=
`(1)/(2)*(9-a)*(4(9-a))/(3)= ( 2(9-a)^(2) )/(3)`

6.

x=y-12


`( 2a^(2) )/(3)=( 2(9-a)^(2) )/(3) - (36)/(3)`


`2a^(2)=2(81-18a+ a^(2) )-36`


`2a^(2)=162-36a+ 2a^(2) -36`


`36a=126`

a=126/36=3.5