Maximum of rectangle in triangle

Question

We have triangle ABC, AB=13cm AC=14cm and BC=15cm. On AC we put a K then AK=x(cm) and we create a rectangle KLMN that is in ABC. Find x for the area of rectangle KLMN is maximum. Sorry for my English writing.

Sorry I cannot post the photo – user343864 Jun 12 '16 at 14:07 — user343864, Jun 12 '16 at 14:07

score 0 · Accepted Answer · answered Jun 12 '16 at 14:18

0

The maximum rectangle area occurs when the midpoints of two of the sides of the triangle were joined to make a side of the rectangle and its area is thus 50% or half of the area of the triangle or 1/4 of the base times height.

Proof : http://mathcountsnotes.blogspot.in/2013/05/the-largest-rectangle-inscribed-in-any.html

Hence, at this point of time we know that x can be 6.5, 7 or 7.5 .

Since , K is on AC , which is part of the rectangle KLMN . Thus ,

x = 7

answered Jun 12 '16 at 14:18

saruftw

141

Sorry can I get the answer using derivative? – user343864 Jun 12 '16 at 14:24
You want steps / proof ? Refer the link . – saruftw Jun 12 '16 at 14:24

Maximum of rectangle in triangle

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