Question

The lengths of the sides of a right angled triangle forms the terms of an arithmetic sequence. If the hypotenuse is 15 cm in length what is the length of the other two sides.​

Answer 1

Answer :

• Lengths of other two sides are 9 cm and 12 cm

Explanation :

given that, hypotenuse is 15 cm

Let, the smallest side of right angled triangle be x

and, other non-hypotenuse side be y

it is also given, that sides of right angled triangle forms an Arithmetic sequence

therefore,

Terms of AP ( in increasing order of values ) will be,

x , y , 15

so,

→ y = ( 15 + x ) / 2   ....equation (1)

Now,

Using Pythagoras theorem in the given right angled triangle

→ ( 15 )² = x² + y²

→ 225 = x² + y²

[ using equation (1) ]

→ 225 = x² + {( 15 + x )/2 }²

→ 225 = x² + ( 225 + x² + 30 x ) / 4

[ multiplying by 4 both sides ]

→ 900 = 4 x² + 225 + x² + 30 x

→ 5 x² + 30 x - 675 = 0

[ dividing by 5 both sides ]

→ x² + 6 x - 135 = 0

→ x² + 15 x - 9 x - 135 = 0

→ x ( x + 15 ) - 9 ( x + 15 ) = 0

→ ( x - 9 ) ( x + 15 ) = 0

so, we get

→ x = 9

[ putting value of x in equation (1) ]

→ y = ( 15 + x ) / 2

→ y = ( 15 + 9 ) / 2

→ y = 12

therefore,

Length of the other two sides of right angled triangle (other than hypotenuse) is 9 cm and 12 cm .

Answer 2

$\bf{\underline{Solution:-}}$

The sides can be represented by a-d, a, a+d where

• a+d = the hypotenuse = 15 cm

So the first equation comes from that:

• a+d = 15

The second equation comes from the Pythagorean theorem:

→ (a-d)² + a² = (a+d)²

→ a² = (a+d)² - (a-d)²

→ a² = [(a+d)-(a-d)][(a+d)+(a-d)]

→ a² = [a+d-a+d][a+d+a-d]

→ a² = [2d][2a]

→ a² = 4ad

→ a² - 4ad = 0

→ a(a - 4d) = 0

→ a = 0; a - 4d = 0

→ a = 4d

• For a = 0, not possible because sides of a triangle are positive numbers.

For a = 4d

→ a+d = 15

→ 4d+d = 15

→ 5d = 15

→ d = 3

→ a = 4(3)

→ a = 12

$\bf{\underline{Hence:-}}$

• The sides are
• a = 12
• a - d = 12 - 3 = 9
• a + d = 12 + 3 = 15 ( given )