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Question: ![](https://www.vedantu.com/question-sets/7e10f02b-4ef8-46a9-ba90-225cb0748aff3220115282937076377.pn...


In the given figure, AB=2,BC=6,AE=6,BF=8,CE=7 and CF=7.AB=2,BC=6,AE=6,BF=8,CE=7\text{ and }CF=7. The ratio of ar ABDE to the arΔCDFar\Delta CDF is:
A. 1:1
B. 2:1
C. 1:2
D. None

Explanation

Solution

To find the ratio of ar ABDE: ar ΔCDF\Delta CDF we need to find the area of a quadrilateral ABDE & area of ΔCDF\Delta CDF. For that first we will observe the given figure, by this we will find that ΔACE&ΔBCF\Delta ACE\And \Delta BCF is congruent. Then by taking Area of Quadrilateral ABDE = Area of ΔACE  Area of ΔBCD\text{ABDE = Area of }\Delta ACE\text{ }-\text{ Area of }\Delta BCD and Area of ΔCDF=Area of ΔBCF Area of ΔBCD\text{Area of }\Delta CDF=\text{Area of }\Delta BCF-\text{ Area of }\Delta BCD . Then we can find the ratio by substituting the values.
Complete step-by-step answer:

By observing the figure, we get to know that AB=2,BC=6,AE=6,BF=8,CE=7,CF=7 and AC=AB+BCAB=2,BC=6,AE=6,BF=8,CE=7,CF=7\text{ and }AC=AB+BC.
By substituting the values of AB and BC, we get –
∃ΑΧ=2+6=8∃
We can also observe from the ΔACE&ΔBCF\Delta ACE\And \Delta BCFthat –
AC=BF=8cm AE=BC=6cm CE=CF=7cm \begin{aligned} & AC=BF=8cm \\\ & AE=BC=6cm \\\ & CE=CF=7cm \\\ \end{aligned}
As the sides of both the triangles are same, we can say that ΔACE&ΔBCF\Delta ACE\And \Delta BCF are congruent.
Therefore, Area of ΔACE= Area of ΔBCF\text{Area of }\Delta ACE=\text{ Area of }\Delta BCF.
Let us consider ‘x’ as the Area of ΔACE=Area of ΔBCF=x\text{Area of }\Delta ACE=\text{Area of }\Delta BCF=x ……………………………… (1)
Now, we need to find the ratio of the area of quadrilateral ABDE to the area of ΔCDF\Delta CDF.
To find this let us draw the figure and shade the area which we need to find.

Here, the shaded regions are quadrilateral ABDE and ΔCDF\Delta CDF.
From the figure we get that –
Area of Quadrilateral ABDE = Area of ΔACE  Area of ΔBCD\text{ABDE = Area of }\Delta ACE\text{ }-\text{ Area of }\Delta BCD
By substituting the value of ΔACE\Delta ACE from equation (1) we get –
Area of ABDE =xΔBCD=x-\Delta BCD …………………….. (2)
We can also observe that –
Area of ΔCDF=Area of ΔBCF Area of ΔBCD\text{Area of }\Delta CDF=\text{Area of }\Delta BCF-\text{ Area of }\Delta BCD.
By substituting the value of ΔACE\Delta ACE from equation (1) we get –
Area of ΔCDF=xΔBCD\text{Area of }\Delta CDF=x-\Delta BCD ……………………….. (3)
Now, we will find the ration of ar ABDE: ar ΔCDF\Delta CDF
Therefore,
Area of Quadrilateral ABDEArea if ΔCDF\dfrac{\text{Area of Quadrilateral ABDE}}{\text{Area if }\Delta \text{CDF}}
By substituting the values from equation (1) and (2) , we get that –
xΔBCDxΔBCD\Rightarrow \dfrac{x-\Delta BCD}{x-\Delta BCD}
By cancelling the common factors from numerator and denominator, we get –
11\Rightarrow \dfrac{1}{1}
Therefore, the ration of ar ABDE: ar ΔCDF\Delta CDF is 1:1
Hence, option A. is the correct answer.

Note: Students should know that, if all the sides of two triangles are proportional, then this is called SSS similarity and the two triangles called congruent or similar triangles.
For example: In our question –
For ΔACE&ΔBCF\Delta ACE\And \Delta BCF
AEBC=CECF=ACBF\dfrac{AE}{BC}=\dfrac{CE}{CF}=\dfrac{AC}{BF}
By substituting their values, we get –
66=77=88\dfrac{6}{6}=\dfrac{7}{7}=\dfrac{8}{8}
By cancelling the common factors from numerator and denominator, we get –
1=1=11=1=1
Therefore, it proves that ΔACEΔBCF\Delta ACE\cong \Delta BCF which is congruent.