Final answer:
To solve the problem, we need to use the Pythagorean theorem and two equations related to the lengths of the sides of the triangle. By solving the equations, we can find the lengths of the sides of the triangle.
Step-by-step explanation:
A right-angled triangle follows the Pythagorean theorem, which states that the sum of the squares of the lengths of its sides is equal to the square of the length of its hypotenuse. So, we can write the equation as:
a2 + b2 = c2
where a and b are the lengths of the two legs of the triangle, and c is the length of the hypotenuse.
In this case, we are given that the sum of squares of the lengths of the three sides is 98cm2. So, we have:
a2 + b2 + c2 = 98
We are also given that the sum of lengths of the three sides is 16cm. So, we have:
a + b + c = 16
With these two equations, we can solve for the lengths of the sides.
Let's represent the length of the two legs as a = x and b = y, and the length of the hypotenuse as c = z.
Using the substitution method, we can solve the equations:
x + y + z = 16 ---- (1)
x2 + y2 + z2 = 98 ---- (2)
Solving equation (1) for z, we get:
z = 16 - x - y
Substituting this value of z in equation (2), we get:
x2 + y2 + (16 - x - y)2 = 98
Expanding and simplifying the equation, we have:
2x2 + 2y2 + 32x + 32y - 160 = 0
Dividing the equation by 2, we get:
x2 + y2 + 16x + 16y - 80 = 0
Completing the square for both x and y, we have:
(x + 8)2 + (y + 8)2 = 160
Now, we have a relationship between the lengths of the two legs, x and y.
We can substitute various values of x and solve for y, or vice versa, to find the lengths of the sides of the right-angled triangle.