Question

if the sides of a right angle triangle in A.p then find the ratio of sides​

Answer 1

Answer:

The sides of a right angled triangle are in Ap find the ratio of their sides

So the ratio of sides is 3:4. The sides of a right triangle are in arithmetic progression.Let one of the legs be x, the other leg x+d, and the hypotenuse

x+2d. ... So the ratio of sides is 3:4 .

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Answer 2

Let the sides be  a, b, c  such that  a < b < c. We have to find a : b : c.

For every right triangle, the hypotenuse is the largest side. Here 'c' is the hypotenuse.

Given that  a, b, c  are in AP. So we can write that,

$2b=a+c\quad\implies\quad c=2b-a\quad\longrightarrow\quad(1)$

Since the triangle is right angled, we can apply Pythagoras' Theorem.

$\begin{aligned}&c^2=a^2+b^2\\\\\implies\ \ &(2b-a)^2=a^2+b^2\quad[\text{From (1)}]\\\\\implies\ \ &4b^2-4ab+a^2=a^2+b^2\\\\\implies\ \ &3b^2-4ab=0\\\\\implies\ \ &3b=4a\\\\\implies\ \ &a:b=3:4\end{aligned}$

Let $a=3x\quad\&\quad b=4x.$ Since he three sides are in AP, $c=5x$ because the common difference is $x$ here. Or Pythagoras' Theorem can be applied too.

Hence the ratio is $\mathbf{3:4:5}.$