The circumference of a right-angled triangle is 24 cm and its hypotenuse is 10 cm. Determine the lengths of the other two sides of the right-angled triangle.
Let the two legs of the right-angled triangle be represented by a and b.
Since the hypotenuse of a right-angled triangle is related to the legs by the Pythagorean theorem, we have:
a^2 + b^2 = c^2
a^2 + b^2 = 10^2
a^2 + b^2 = 100
The circumference of the triangle is given as 24 cm, therefore:
a + b + c = 24
a + b + 10 = 24
a + b = 14
Now we have a system of two equations:
a^2 + b^2 = 100
a + b = 14
There are different methods to solve this system of equations, through substitution or elimination methods. One way to find the solution is by using the elimination method.
From equation 2, we have:
a = 14 - b
Substitute this value of a into equation 1:
(14 - b)^2 + b^2 = 100
196 - 28b + b^2 + b^2 = 100
2b^2 - 28b + 96 = 0
b^2 - 14b + 48 = 0
By solving the quadratic equation using the quadratic formula:
b = [14 ± sqrt(196 - 192)] / 2
b = [14 ± 2] / 2
b = 8 or b = 6
Therefore, the lengths of the other sides of the right-angled triangle are 6 cm and 8 cm.