Are They In Progression?

Geometry Level 1

The sides of a right triangle are $a$ , $a+d$ and $a+2d$ with $a$ and $d$ both positive. What is the value of $a:d$ ?

Note : Image drawn not up to scale.

2:1

3:1

1:4

1:3

2 solutions

Alan Enrique Ontiveros Salazar
Oct 3, 2015

Relevant wiki: Pythagorean Theorem

Because $a<a+2d$ and $a+d<a+2d$ . Then, $a+2d$ must be the longest side of the triangle, and because it is a right angle triangle, $a+2d$ is the hypothenuse of this triangle.

By the Pythagorean theorem , we know that $(a)^2+(a+d)^2=(a+2d)^2$ , expand to get:

$a^2+a^2+2ad+d^2=a^2+4ad+4d^2$

$a^2-2ad-3d^2=0$

$(a+d)(a-3d)=0$

$a=3d$

Hence, $a:d=3:1$ .

I like the way you found the zeros. I never thought of doing that before with variables. I know about factoring, but the substitution for a => 3d is clever

Deep state of denial Burgert - 1 year, 9 months ago

Or set d=1 and solve a^2 - 2a - 3 = 0.

Jonas Brinkmann - 4 years, 11 months ago

Shantanu Saxena
Mar 8, 2017

* * Note that sides of the triangle are in A.P, so one of the most famous right triangle i.e $(5,4,3)$ triangle with $a=3,d=1$

Are They In Progression?

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