Question

how we van find question mark (?) in this shape
http://cdn.persiangig.com/download/8V3100/dl

Answer 1

Answer on Question #57487 – Math – Geometry

Question

How we can find question mark (?) in this shape:

Solution

Let DN=x. Triangles BEM and CEK are similar, because their corresponding angles are equal.

Following this we have:

$$\frac{BM}{CK} = \frac{BE}{CE} = \frac{5}{7}. \quad \text{Let} \quad \frac{BE}{CE} = \frac{5y}{7y}.$$ $$BC = BE + EC = 5y + 7y = 12y.$$

Then AD=BC (as the same opposite sides of the rectangle), hence AD=12y.

Triangles BEM and DNA are similar, because their corresponding angles are equal too.

Following this we have:

$$\frac{BE}{BM} = \frac{DN}{DA}. \quad \frac{5y}{5} = \frac{x}{12y}. \quad y = \sqrt{\frac{x}{12}}. \quad AD = 12y = 12\sqrt{\frac{x}{12}} = \sqrt{12x}.$$

By Pythagorean theorem in triangle AND, AN=$\sqrt{ND^2 - AD^2} = \sqrt{x^2 - 12x}$.

Triangles NAF and DNA are similar, because their corresponding angles are equal too.

Following this we have:

$$\frac {D N}{A N} = \frac {A N}{A F},$$ $$\frac {x}{\sqrt {x ^ {2} - 1 2 x}} = \frac {\sqrt {x ^ {2} - 1 2 x}}{4}.$$ $$x ^ {2} - 1 2 x = 4 x.$$ $$x ^ {2} - 1 6 x = 0.$$ $$x = 1 6.$$

Answer: 16.

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