Question

a) Lengths of sides of a right angled triangle are in arithmetic sequence with common difference d. If the length of the smallest side of the triangle is x - d, write the length of its other two sides.
b) Show that any right angled triangle with sides in arithmetic sequence is similar to the right angled triangle with sides 3, 4 and 5.

Answer 1

a) Length of the smallest side is x - d.
Let the length of the other two sides is x, x + d.
Using Pythagoras Theorem,
$(x+d{)}^2=x^2+(x-d{)}^2$
$x^2+2xd+d^2=x^2+x^2-2xd+d^2$
$x^2=4xd$
$x=4d$
Hence length of the sides is,
$x-d=4d-d=3d$
$x=4d$
$x+d=4d+d=5d$
b) Let $x-d,x,x+d$ be the sides of a right angled triangle in arithmetic progression. Using Pythagoras Theorem,
$(x+d{)}^2=x^2+(x-d{)}^2$
$x^2+2xd+d^2=x^2+x^2-2xd+d^2$
$x^2=4xd$
$x=4d$
Hence length of the sides is,
$x-d=4d-d=3d$
$x=4d$
$x+d=4d+d=5d$
Hence the sides 3d, 4d, 5d are similar to pythagorean triplet sieds 3, 4, 5.