if three sides of a right angled triangle is given by (x+3)cm,(x+1)cm and 2xcm as the hypotenuse find the value of x
Pythagorean theorem:
( x + 3 )² + ( x + 1 )² = ( 2 x )²
x² + 2 ∙ x ∙ 3 + 3² + x² + 2 ∙ x ∙ 1 + 1² = 4 x²
x² + 6 x + 9 + x² + 2 x + 1 = 4 x²
2 x² + 8 x + 10 = 4 x²
Subtract 4 x² to both sides
2 x² + 8 x + 10 - 4 x² = 4 x² - 4 x²
- 2 x² + 8 x + 10 = 0
Divide both sides by - 2
x² - 4 x - 5 = 0
The solutions are x = 5 and x = - 1
Length cannot be negative so x = 5
Proof:
( x + 3 )² + ( x + 1 )² = ( 2 x )²
( 5 + 3 )² + ( 5 + 1 )² = ( 2 ∙ 5 )²
8² + 6² = 10²
64 + 36 = 100
100 = 100
To find the value of x in a right-angled triangle, we'll use the Pythagorean Theorem, which states that the square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the lengths of the other two sides.
In this case, we have a hypotenuse of length 2x cm and two other sides of lengths (x+3) cm and (x+1) cm.
Applying the Pythagorean Theorem, we can write the equation as:
(2x)² = (x+3)² + (x+1)²
Expanding the equation, we get:
4x² = (x+3)(x+3) + (x+1)(x+1)
Simplifying further:
4x² = (x² + 6x + 9) + (x² + 2x + 1)
Grouping like terms:
4x² = 2x² + 8x + 10
Rearranging the equation to set it equal to zero:
2x² + 8x + 10 - 4x² = 0
Combining like terms:
-2x² + 8x + 10 = 0
To solve this quadratic equation, we can use factoring or the quadratic formula. Let's use the quadratic formula:
x = (-b ± √(b² - 4ac)) / 2a
For our equation, a = -2, b = 8, and c = 10. Substituting these values into the formula:
x = (-(8) ± √((8)² - 4(-2)(10))) / (2(-2))
Simplifying further:
x = (-8 ± √(64 + 80)) / (-4)
x = (-8 ± √(144)) / (-4)
x = (-8 ± 12) / (-4)
We have two possible values of x:
1. x = (-8 + 12) / (-4) = 4 / (-4) = -1
2. x = (-8 - 12) / (-4) = -20 / (-4) = 5
Therefore, the value of x can be either -1 or 5.
To find the value of x, we can use the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
So, applying the Pythagorean theorem, we can write:
(Adjacent side)^2 + (Opposite side)^2 = (Hypotenuse)^2
Substituting the given values, we have:
(x+3)^2 + (x+1)^2 = (2x)^2
Expanding and simplifying, we get:
x^2 + 6x + 9 + x^2 + 2x + 1 = 4x^2
Combining like terms, we have:
2x^2 + 8x + 10 = 4x^2
Moving all terms to one side, we get:
0 = 2x^2 - 8x - 10
To solve this quadratic equation, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
For our equation, a = 2, b = -8, and c = -10.
Substituting these values into the quadratic formula, we have:
x = (-(-8) ± √((-8)^2 - 4(2)(-10))) / 2(2)
Simplifying further, we get:
x = (8 ± √(64 + 80)) / 4
x = (8 ± √144) / 4
x = (8 ± 12) / 4
There are two possible solutions:
1. x = (8 + 12) / 4 = 5
2. x = (8 - 12) / 4 = -1/2
So, the two possible values of x are 5 and -1/2.