In the figure, if $$AE = 3$$, $$DE = 5$$, and $$ CE = 7$$, then $$BF$$ equals:
![Dandelin Spheres Theory - Math Images](uploads/posts/2022/08/medium/d851dd4af63cca7c7dfa25c66ff51bd2.png)(275)(223)(264)

Question

In the figure, if $$AE = 3$$, $$DE = 5$$, and $$ CE = 7$$, then $$BF$$ equals: 
![Dandelin Spheres Theory - Math Images](uploads/posts/2022/08/medium/d851dd4af63cca7c7dfa25c66ff51bd2.png)(275)(223)(264)

Uniteasy Admin · Accepted Answer

$$\angle BAF$$ and $$\angle DAE$$ are complementary angles, which means their sum is 90$$\degree  $$. Therefore, $$ 	riangle AED$$ and $$	riangle$$ BFA are similar. Similarly, $$	riangle CFB$$ and $$	riangle DEC$$ are also similar.
Therefore, we get the following equations.
$$\dfrac{BF}{AF} = \dfrac{AE}{DE} =\dfrac{3}{5}   $$n
$$\dfrac{BF}{CF} = \dfrac{CE}{DE} = \dfrac{7}{5}      $$n
Divide (1) by (2), we get
$$\dfrac{CF}{AF} = \dfrac{3}{7} $$n   
On the other hand,
$$CF+AF = CE+AE = 10$$n
Solve the system of equations (3) and (4),
We get, $$CF = 3$$ and $$AF = 7$$
Using the equation (2),
$$BF =  \dfrac{7}{5}	imes   CF =   \dfrac{7}{5}	imes 3 = 4.2$$

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