In a right angled angel,two sides enclosing the right angle measure (3x-2)cm and (x+2)cm.If the area of the triangle is 17.5 cm,find the length of these two sides
To find the length of the two sides of a right-angled triangle, we can use the Pythagorean theorem which states that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
Let (3x-2) represent the length of one side and (x+2) represent the length of the other side.
According to the Pythagorean theorem:
Hypotenuse^2 = (3x-2)^2 + (x+2)^2
Given that the area of the triangle is 17.5 cm, we know:
Area = (1/2) base * height
17.5 = (1/2)(3x-2)(x+2)
Now we can solve for x. Multiplying both sides of the equation by 2 gives:
35 = (3x-2)(x+2)
Expanding the right side gives:
35 = 3x^2 + 6x - 2x - 4
Combining like terms, we have:
0 = 3x^2 + 4x - 39
To solve this quadratic equation, we can factor or use the quadratic formula. Factoring the equation does not yield neat integer roots, therefore we will use the quadratic formula:
x = [-b ± √(b² - 4ac)] / (2a)
In this case, a = 3, b = 4, and c = -39. Substituting these values into the quadratic formula gives:
x = [-4 ± √(4² - 4(3)(-39))] / (2(3))
x = [-4 ± √(16 + 468)] / 6
x = [-4 ± √484] / 6
x = [-4 ± 22] / 6
Simplifying further, we have:
x = (18 / 6) or x = (-26 / 6)
x = 3 or x = -13/3
We discard the negative value of x since length cannot be negative.
Therefore, x = 3.
Now, we can substitute x = 3 back into the original expressions for the lengths of the sides:
Length of one side = 3x - 2 = (3*3) - 2 = 7 cm
Length of the other side = x + 2 = 3 + 2 = 5 cm
Therefore, the lengths of the sides of the right-angled triangle are 7 cm and 5 cm.
To find the lengths of the two sides in a right-angled triangle, we can use the Pythagorean theorem, which states that in a right 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.
Let's label the sides as follows:
Hypotenuse = (3x - 2) cm
Other side = (x + 2) cm
According to the Pythagorean theorem, we have:
(Hypotenuse)^2 = (Other side)^2 + (Other side)^2
Simplifying the equation:
(3x - 2)^2 = (x + 2)^2 + (x + 2)^2
Expanding and simplifying further:
9x^2 - 12x + 4 = x^2 + 4x + 4 + x^2 + 4x + 4
Combining like terms:
9x^2 - 12x + 4 = 2x^2 + 8x + 8
Rearranging the equation:
9x^2 - 2x^2 - 12x - 8x + 4 - 8 = 0
Combining like terms again:
7x^2 - 20x - 4 = 0
Now, to solve the quadratic equation, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 7, b = -20, and c = -4.
Calculating the values:
x = (-(-20) ± √((-20)^2 - 4 * 7 * -4)) / (2 * 7)
x = (20 ± √(400 + 112)) / 14
x = (20 ± √512) / 14
x = (20 ± 22.63) / 14
Now calculating the two possible values of x:
x₁ = (20 + 22.63) / 14 ≈ 3.9
x₂ = (20 - 22.63) / 14 ≈ -0.9
Since lengths cannot be negative, we can disregard the second solution.
Therefore, the length of the sides are:
Hypotenuse = (3x - 2) cm = (3 * 3.9 - 2) cm ≈ 9.7 cm
Other side = (x + 2) cm = (3.9 + 2) cm = 5.9 cm
Hence, the length of the hypotenuse is approximately 9.7 cm, and the length of the other side is approximately 5.9 cm.