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Try this beautiful problem from Geometry based on Side Length of Rectangle.

Rectangle $A B C D$ has $A B=4$ and $B C=3 .$ Segment $E F$ is constructed through $B$ so that $E F$ isperpendicular to $D B$, and $A$ and $C$ lie on $D E$ and $D F$, respectively. What is $E F$ ?

- $9$
- $10$
- $\frac{125}{12}$
- \(\frac{103}{9}\)
- \(12\)

Triangle

Rectangle

Geometry

But try the problem first...

Answer: $\frac{125}{12}$

Source

Suggested Reading

AMC-10A (2009) Problem 10

Pre College Mathematics

First hint

We have to find out the length of \(EF\)

Now $BD$ is the altitude from $B$ to $EF$, we can use the equation $BD^2 = EB\cdot BF$. ( as \(\triangle BDE \sim \triangle BDF\)).so we have to find out \(BE\) and \(BF\)

Can you now finish the problem ..........

Second Hint

Now Clearly, $\triangle BDE \sim \triangle DCB$. Because of this, $\frac{A B}{C B}=\frac{E B}{D B}$. From the given information and the Pythagorean theorem, $A B=4, C B=3$, and $D B=5 .$ Solving gives $E B=20 / 3$

We can use the above formula to solve for $B F . B D^{2}=20 / 3 \cdot B F$. Solve to obtain $B F=15 / 4$

can you finish the problem........

Final Step

Therefore $E F=E B+B F=\frac{20}{3}+\frac{15}{4}=\frac{80+45}{12}$

- https://www.cheenta.com/problem-from-probability-amc-8-2004problem-no-21/
- https://www.youtube.com/watch?v=VLyrlx2DWdA&t=15s

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This problem can be done using coordinate geometry by using convenient coordinate system. Let us consider $D$ to be the origin. Then by your diagram it follows that the coordinate of $A,B$ and $C$ are respectively $(-0,-3),(4,-3)$ and $(4,0).$ Then the equation of the line joining $B$ and $D$ is $3x+4y=0.$ So the equation of the line $L$ passing through $B$ and perpendicular to $BD$ is given by $4x - 3y = 25.$ So coordinate of $F$ and $E$ can be found by the point of intersections of $L$ with $X$-axis and the point of intersection of $L$ with $Y$-axis respectively. Therefore the coordinate of $E$ and $F$ are respectively $\left (0, -\frac {25} {3} \right)$ and $\left (\frac {25} {4} , 0 \right ).$ So the length of $EF$ is $$\sqrt {\left (-\frac {25} {4} \right )^2 + \left (- \frac {25} {3} \right )^2} = \frac {125} {12}.$$