Question

The sides of a right angled triangle are in A.P. The ratio of sides is
A) 1 : 2 : 3
B) 2 : 3 : 4
C) 3 : 4 : 5
D) 5 : 8 : 3

Answer 1

A.P. stands for Arithmetic progressions. Arithmetic Progression is the sequence having equal difference between two adjacent terms.
Example 1, 3, 5 , 7,9………………..
Hence 3 – 1 = 2
5 – 3 = 2
7 – 5 = 2
9 – 7 = 2
In the above sequence, the common difference between the adjacent terms is 2. Hence, the sequence is A.P. Also, in the right angled triangle, we can use the Pythagoras property i.e. $\;hypotenuse{\;^2} = bas{e^2} + perpendicular{r^2}$ . where hypotenuse is the longest side.

Complete step by step solution:
Step 1
If we consider a right angled triangle Δ ABC and ∠ B $= {90^o}$ then we know that AC would be the hypotenuse. So, on applying Pythagoras property;

i.e. ${H^2} = {B^2} + {P^2}$
⇒ $A{C^2} = A{B^2} + B{C^2}$

Step2
Also it is given that the length of the sides of the triangle are in A.P. so, suppose the sides of the given triangle are $x + d$ , $x + 2d$ , $x + 3d$ ( where d is the common difference.)
Step3.
We also know that hypotenuse is the longest side. So, consider $x + 3d$ as the hypotenuse and the rest two sides are base and perpendicular.
Step4.
Using Pythagoras property as mentioned above
${H^2} = {B^2} + {P^2}$
⇒ ${(x + 3d)^2} = {(x + d)^2} + {(x + 2d)^2}$
On using identity, we get;
${x^2} + 9{d^2} + 6xd = {x^2} + {d^2} + 2xd + {x^2} + 4{d^2} + 4xd$
$\Rightarrow {x^2} + 9{d^2} + 6xd = 2{x^2} + 5{d^2} + 6xd$
On rearranging the terms and performing the required operations;
${x^2} + 9{d^2} + 6xd - (2{x^2} + 5{d^2} + 6xd) = 0$
$\Rightarrow {x^2} - 4{d^2} = 0$
$\Rightarrow {x^2} = 4{d^2}$
Step 5.
On taking the square root on the expression on both the sides :
${({x^2})^{\frac{1}{2}}} = {(4{d^2})^{\frac{1}{2}}}$
⇒ x = 2d
Step 6.
Substituting the value of x = 2d in the value of sides of the triangle.
$x + d = 2d + d = 3d$
$x + 2d = 2d + 2d = 4d$
$x + 3d = 2d + 3d = 5d$

Hence, the sides of the triangle are in the ratio 3 : 4 : 5.

Note:
The arithmetic progression are the sequences in which the terms are separated with the equal difference in all the adjacent terms. If the first term of A.P. is represented by “a” and the common difference between them is represented by “d” . then, the nth term would be represented by ${a_n}$ and calculated as
${a_n} = a + (n - 1)d$ where n is the number of terms, we want to calculate.
In case of right angle triangle, the Pythagoras property given by the great Mathematician Pythagoras is commonly used when the discussion is all about its sides..
In step 6 , the sides came out to be 3d , 4d, and 5d . so to write in the proportion, we needed to omit the proportional constant i.e. d here.
Hence, option(c) is correct i.e. 3 : 4 : 5.