GATE Electrical 2024 | GA | Question: 7

Question

​​​​In the following figure,
\[
\begin{array}{c}
\mathrm{CD}=5 \mathrm{~cm}, \mathrm{BE}=10 \mathrm{~cm}, \mathrm{AE}=12 \mathrm{~cm}, \\
\angle \mathrm{DAB}=\angle \mathrm{DCB} \text {, and } \angle \mathrm{DAE}=\angle \mathrm{DBC}=90^{\circ}
\end{array}
\]

Points $\text{AFCD}$ create a rhombus.

The length of $\mathrm{BF}$ (in $\mathrm{cm}$ ) is

• A. $3$
• B. $2$
• C. $4$
• D. $6$

Answer 1

Let's denote \( BF \) as \( x \).

Given: \( \angle DAB = \angle DCB \).

From this, we deduce that \( AD = AC \) (because \( \triangle DAB \) and \( \triangle DCB \) are isosceles triangles sharing side \( AD \) and \( AC \)).

Additionally, in a rhombus, the diagonals bisect each other at right angles. Therefore, \( DB = FB \).

We also know the Pythagorean theorem applies to the right triangle \( \triangle ADE \):

\[
AD^2 + AE^2 = ED^2
\]

Plugging in the given values, we have:

\[
AD = 5 \quad \text{and} \quad AE = 12
\]

So,

\[
5^2 + 12^2 = ED^2
\]

\[
25 + 144 = (10 + x)^2
\]

\[
169 = (10 + x)^2
\]

Taking the square root of both sides:

\[
\sqrt{169} = \sqrt{(10 + x)^2}
\]

\[
13 = 10 + x
\]

Solving for \( x \):

\[
x = 13 - 10
\]

\[
x = 3
\]

Therefore, the value of \( x \) is 3.

Comments

@Stuti7 you have obtained the value of $DF = 3cm$ but in the question they are asking about $BF = ?$. Here, $BF = FD / 2 = 1.5cm$ and there is no matching option. That's why this question itself is wrong.

Answer 2

$AFCD$ is rhombus(four sides and opposite angles are euqal) that means $CD = DA = AF = FC = 5cm$

 

given $\angle DAE = 90^{\circ}$ and $AE = 12cm, \\\ CD = 5cm$

so, $DE^2 = AE^2 + AD^2 \rightarrow DE^2 = 12^2 + 5^2 = 13^2$

 

given $BE = 10cm$

by the geometry we can infer that $DB = DE - BE \rightarrow DB = 13 - 10 = 3cm$

 

given $\angle DBC = 90^{\circ}$

by this $DC^2 = DB^2 + BC^2 \rightarrow 5^2 = 3^2 + BC^2 \rightarrow BC^2 = 4^2$

 

now, $FC^2 = BC^2  + BF^2 \rightarrow  5^2 = 4^2 + BF^2 \rightarrow BF^2 = 3^2 $

$BF = 3cm$

Answer 3

From the above methods the value of BF is 3.

But, lets try another method (The answer should be same right?).
Given that DBC = 90.

Therefore, ABF = 90, (opposite angles).

Hence, triangle ABF and ABE are right angled.

Now, AE = 12, BE = 10
Therefore, AB^2 = 12*12 - 10*10 = 44

Thus, AB = sqrt(44) = 6.63

In triangle ABF,

AB is known, and AF = CD = 5 (as AFCD is a rhombus)
This means that the hypotenuse of the triangle is smaller than one of its side which is not possible.
Hence the question is ambiguous.
 

Comments

agreed with your logic length of diagonal can not be more than the sum of two adjecent sides.

Answer 4

AD^2  + AE^2 = DE^2

AD=CD=5, AE=12, DE=BE+BD

5^2+12^2 = DE^2

25+144=DE^2

169=DE^2

DE=13 = DB+BE = 10 + DB

DB = BF = 1310=3

Answer is 3   i.e. A