Question

three sides of a right triangle are three consecutive terms of an arithmetic sequence of common different 4. if x-4 is the length of smallest side, find length of other two side. Find length of sides of the triangle ​

Answer 1

Given :

• Sides of a right angled triangle are three consecutive terms of an Arithmetic sequence with common difference 4.

• x-4 is the length of smallest side

To Find :

• The length of sides of the triangle.

Solution :

Let the three sides of the right angled triangle which are consecutive terms of the arithmetoc progression be " x - d " , " x " and " x + d ".

Where d is common difference.

In this case we are given that common difference as 4. x - 4 , x , x + 4 now becomes the sides of right angled triangle (as per given information).

We are given that x - 4 is the smallest side. x + 4 becomes the largest side (since it has higher length compared to other two sides)

In a right angled triangle hypotenuse is the largest side. So , In this case x + 4 is the hypotenuse of the given right angled triangle. Other two sides are x and x - 4.

Now , using pythogorean theorem ;

$\\ \star \: \boxed{\purple{\sf{(hypotenuse) ^{2} = ({side}_1) {}^{2} + ({side}_2) {}^{2} }}}$

Now , by substituting the values we have ;

$\\ : \implies \sf \: (x + 4) {}^{2} = {(x)}^{2} + {(x - 4)}^{2}$

Now using the identities ,

• (a+b)² = a² + b² + 2ab
• (a-b)² = a² + b² - 2ab

we get ;

$\\ : \implies \sf \: {x}^{2} + 16 + 8x = {x}^{2} + {x}^{2} + 16 - 8x$

Cancelling x² + 16 on both sides of the equation ;

$\\ : \implies \sf \: 8x = {x}^{2} - 8x$

$\\ : \implies \sf 16x = {x}^{2}$

$\\ : \implies \sf {\boxed{\pink {\sf{16 = x}}}} \: \bigstar$

Now ;

$\\ \longmapsto \sf \: x - 4 = 16 - 4 = 12$

$\longmapsto \sf \: x = 16$

$\longmapsto \sf \: x + 4 = 16 + 4 = 20$

Hence , The sides of the triangle are 12 , 16 and 20.

Answer 2

Explanation:

Given :

• three sides of a right triangle are three consecutive terms of an arithmetic sequence of common different 4.

• if x-4 is the length of smallest side, find length of other two side.

To Find :

• Find length of sides of the triangle

Solution :

Concept :

• Pythagoras theorem states that “In a right-angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides“.

• The sides of this triangles have been named as Perpendicular, Base and Hypotenuse.

• Here, the hypotenuse is the longest side, as it is opposite to the angle 90°.

_____________________________

• According to the Question :

Let The first side is x and x - 4 is the second side and length of the largest side = x + 4

$: \implies \: \: \: \boxed{ \sf \: {Hypotenuse}^{2} = side_1{^{2}} + side_2{^{2}} \: }$

• Substitute all values :

$: \implies \: \: \: \sf \: ({x + 4})^{2} = x{^{2}} + (x - 4){^{2}} \: \\ \\ \\ : \implies \: \: \: \sf \: \cancel{{x}^{2}} + 8x + \cancel{16} = \cancel{ {x}^{2} } + {x}^{2} - 8x + \cancel{16 }\\ \\ \\ : \implies \: \: \: \sf \:8x + 8x = {x}^{2} \\ \\ \\: \implies \: \: \: \sf \:16x = {x}^{2} \\ \\ \\ :\implies \: \: \: \sf \:x = \cancel{ \frac{16x}{x} } \\ \\ \\ :\implies \: \: \: \sf \:x = 16$

Find length of other two side.

• First Length of side x = 16

• Second Length of side x - 4 = 16 - 4 = 12

• Third Length of side x + 4 = 16 + 4 = 20