Question

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

Answer 1

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.