Question

If the length of sides of a right angled triangle are in ap then find their ratio

Answer 1

As the sides of the triangle are in an AP, let the sides be a,  a + d and a + 2d.

Assume that the AP in which the length of these three sides belong is in increasing order, such that a + 2d will be the greatest among the length of the sides.

According to this, side of length a + 2d will be the hypotenuse.

Using Pythagoras' theorem,

$a^2+(a+d)^2=(a+2d)^2 \\ \\ \\ a^2+a^2+2ad+d^2=a^2+4ad+4d^2 \\ \\ \\ a^2+2ad+d^2=4ad+4d^2 \\ \\ \\ a^2+2ad+d^2-4ad-4d^2=0 \\ \\ \\ a^2-2ad-3d^2=0 \\ \\ \\ a^2+ad-3ad-3d^2=0 \\ \\ \\ a(a+d)-3d(a+d)=0 \\ \\ \\ (a+d)(a-3d)=0$

As  a + d  is the length of a side of the triangle,  a + d = 0  is not possible.

$\begin{aligned}\therefore\ \ & a-3d=0 \\ \\ &a=3d\end{aligned}$

Now, writing the ratio of the sides,

$a\ :\ a+d\ :\ a+2d \\ \\ \\ 3d\ :\ 3d+d\ :\ 3d + 2d \\ \\ \\ 3d\ :\ 4d\ :\ 5d \\ \\ \\ \large \bold{3\ :\ 4\ :\ 5}$

Thus the sides are in the ratio 3 : 4 : 5.

Answer 2

Given :- The sides of a right angled ∆ are in AP

» We know that to be in AP the numbers must follow a certain patern

» so to find nth term in the series we use

án = a+( n- 1)d

For n= 1

án = a + (1-1)d

= a

For n = 2

án = a+ (2-1)d

= a+d

For n = 3

án = a+ (3-1) d

= a + 2d

•°• The 3 sides of the ∆ be a ,a+d , a+2d ---------(1)

Question :- To find their ratios.

» So to find their ratios we should know their sides .

Let's find

By Pythagoras theorem

(AC)^2 = (AB)^2 + (BC)^2

Here

(a+2d)^2 = (a+d)^2 + (a)^2

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

a^2 + 4d^2 + 4ad -a^2 -d^2 -2ad -a^2 =0

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

a^2 -2ad -3d^2

By Factorization :-

[ Product of 1st and last term in equation = 1× - 3 = - 3 when we add or subtract it multiples we must get the coefficient of middle term » 1-3 = -2 ]

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

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

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

(|) » a- 3d = 0

» a = 3d

(||) a = -d

Now substitute "a" value in the values in equation (1)

a, a+d,a+ 2d

For a = 3d

we get

3d , 3d +d , 3d +2d

3d , 4d , 5d

3, 4 , 5

Their Ratios Will be. 3:4:5 .

For a = -d

-d , -d +d , -d +2d

-d , 0 , d

ẞSo a side Will become zero , Hence this will never happen.

so the answer is

3:4:5

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Glad if helped.