in the given if AB=2,BC=6,AE=6,BF=8,CE=7,CF=7,compute the ratio of the area of quadrilateral ABDE to the area of ∆CDF
Answers
In this diagram we have the following triangles whose area we can find as follows :
Triangle BCD which is similar to ACE.
And BCF
To gets lengths BD and CD for triangle BCD we will use linear scale factor.
We get lsf from lengths AC and BC as follows :
8/6 = 4/3
8(BF) /BD = 4/3
24 = 4BD
BD = 24/4 = 6
CE/CD = 8/6
7/(CD) = 8/6
42 = 8CD
CD = 42/8
= 5.25
The area of ACE :
We get this by herons formula.
Herons formulae is :
A = √S(S - a) (S - b) (S - c)
S = (a + b + c) /2
S = (6 + 7 + 8)/2
= 10.5
A = √10.5(10.5 - 6)(10.5 - 7)(10.5 - 8) = √413.4375 = 20.33
Area of BCF is also equal to 20.33
Area of BCD :
S = (6 + 6+, 5.25) /2 = 8.625
A = √8.625(8.625 - 6)(8.625 - 6)(8.625 - 5.25) = √200.58 = 14.16
= 14.16
Area of CDF :
S = (5.25 + 7 + 2) / 2 = 7.125
A = √7.125(7.125 - 5.25) (7.125 - 7)(7.125 - 2) = √8.558 = 2.925
= 2.925
Area of ABDE.
20.33 - 14.16 = 6.17
The ratio of ABDE : CDF
6.17 : 2.925
Approximately :
6 : 3
= 2 : 1
Triangle BCD which is similar to ACE.
And BCF
To gets lengths BD and CD for triangle BCD we will use linear scale factor.
Here is the LSF
8/6 = 4/3
8(BF) /BD = 4/3
24 = 4BD
BD = 24/4 = 6
CE/CD = 8/6
7/(CD) = 8/6
42 = 8CD
CD = 42/8
= 5.25
The area of ACE :
Here is the formula:
A = √S(S - a) (S - b) (S - c)
S = (a + b + c) /2
S = (6 + 7 + 8)/2
= 10.5
A = √10.5(10.5 - 6)(10.5 - 7)(10.5 - 8) = √413.4375 = 20.33
Area of BCF is also equal to 20.33
Area of BCD :
S = (6 + 6+, 5.25) /2 = 8.625
A = √8.625(8.625 - 6)(8.625 - 6)(8.625 - 5.25) = √200.58 = 14.16
= 14.16
Area of CDF :
S = (5.25 + 7 + 2) / 2 = 7.125
A = √7.125(7.125 - 5.25) (7.125 - 7)(7.125 - 2) = √8.558 = 2.925
= 2.925
Area of ABDE.
20.33 - 14.16 = 6.17
The ratio of ABDE : CDF
6.17 : 2.925
Approximately :
6 : 3
= 2 : 1