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a square as shown in the figure is
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inscribed in a triangle with sides 13 14
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and 15 units find the area of the square
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as usual before watching press pause and
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try to solve the problem yourself there
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are different ways to solve it and we
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will explore one of the quickest methods
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first using Heron's formula we find the
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area of the triangle where p is the
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semi- perimeter of the triangle and a B
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and C are its sides let's label the
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vertices of the triangle and draw the
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height BD from vertex x b knowing the
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length of the triangle's base AC we can
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find the height from the area using the
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right triangles ABD and BDC we apply the
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Pythagorean theorem to find the legs a d
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and DC we label the points where the
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height b d intersects the sides of the
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triangle and the side of the square as a
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result we obtain two pairs of similar
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triangles triangle BF is similar to
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Triangle B A and triangle BGE is similar
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to Triangle BCD from the proportionality
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and corresponding sides of the similar
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triangles we have be relates to Fe as BD
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relates to a d substituting we get 12
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fths let be be 12x then Fe will be 5x
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from the second pair of similar
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triangles be relates to EG as BD to DC
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substituting we get 12 9ths since be is
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12x EG will be 9x e d equals the side of
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the square FG which in turn equals the
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sum of Fe and EG substituting we get 14x
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the height BD equal equals the sum of be
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and EG substituting we get 26x the
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height b d is 12 meaning 26x equal 12
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from which X = 1226 or 613 the side of
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the square FG is equal to 14x
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substituting X we find the side of the
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square the area of the square is equal
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to the square of its side substituting
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we find the area to be
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756 over 169 or this would be 41 and 127
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169 we write the answer the area of the
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square is 41 and 127 over 169 squ units
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the problem is solved if you understood
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the solution give a like and don't
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forget to subscribe to the channel
