Question

The sum of the hypotenuse and the altitude of a right angled triangle dropped on the hypotenuse exceeds the half-perimeter of the triangle.

• A. True
• B. False

Answer 1

Answer: (A)

Let p be the length of perpendicular dropped from the right angle on the
hypotenuse of length c of a right angled triangle of sides a and b. Then
pc = ab = (twice the area of $\Delta$)
Also $a^2+b^2=c^2\dots \dots \left(2\right)$
Now we have to prove $p+c>\frac{a+b+c}{2}$
or $2p>a+b-c$
or $2.\frac{ab}{c}>a+b-c$, by (1)
or $2ab>a^c+bc-c^2=ac+bc-a^2-b^2$ by (2)
or $a^2+b^2+2ab>c(a+b)$
or $(a+b{)}^2>c(a+b)$
or $a+b>c$ $[\because a+b>0]$
which is true since the sum of two sides of a triangle is greater than the third side.