SOLUTION: A farmer has a field in the shape of right angled triangle with legs (sides other than the hypotenuse) of lengths 16 m and 8 m. She wants to leave a space in the form of a square

Question 986115: A farmer has a field in the shape of right angled triangle with legs (sides other than the hypotenuse) of lengths 16 m and 8 m. She wants to leave a space in the form of a square of largest area for growing wheat and remaining area for growing vegetables. Find the length of the side of such a square

Answer by ikleyn(52781) (Show Source): You can put this solution on YOUR website!

We are given a right-angled triangle ABC (see the Figure) with the legs |AB| = 8 and |AC| = 16.
A square ADEF is inscribed to the triangle ABC in such a way that the triangle and
the square have the common right angle LA.
We need to find the length of the square side.

Let x be the square side length.
Then the length of the segment DB is 8 - x and the length of the segment CF is 16 - x.

The triangles DBE and FEC are right angled triangles.
They have congruent acute angles LFCE and LDEB as these angles are the corresponding
angles at the parallel lines AC and DE and the transverse BC.

Hence, the triangles DBE and FEC are similar right-angled triangles.
Therefore, their legs are proportional:

Figure

= ,     or     = .                        (3)

Now we need to solve the last equation for the unknown x. Simplify the equation (3) step by step:

= ,

= ,

= ,

= ,    = = = .

Answer. The side measure of the square is of    m long.

For a similar problem see the lesson Miscellaneous problems on similar triangles in this site.

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