The hypotenuse of a right angled triangle is 65cm. One side is 10 cm shorter than twice the other side. Find the lengths of the 2 sides
(provide a little of steps if available)
The Pythagorean theorem tells you that
65^2 = x^2 + (2x -10)^2
4225 = 5x^2 -40x + 100
Now collect terms
5x^2 -40x -4125 = 0
Divide by 5 to simplify.
x^2 -8x - 825 = 0
That equation can be factored.
(x+25)(x-33) = 0
The only positive root is x = 33 cm
The other side must be 56 cm long
To find the lengths of the two sides of the right-angled triangle, we can use the Pythagorean theorem. The Pythagorean theorem 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's assume that one side of the triangle is "x" cm. According to the problem, the other side is 10 cm shorter than twice the length of the first side. So, the length of the other side can be expressed as "2x - 10" cm.
Now, we can write the equation using the Pythagorean theorem:
(2x - 10)^2 + x^2 = 65^2
Let's solve this equation for "x".
Expanding (2x - 10)^2:
4x^2 - 40x + 100 + x^2 = 65^2
Combining like terms:
5x^2 - 40x + 100 = 65^2
Subtracting 65^2 from both sides:
5x^2 - 40x + 100 - 65^2 = 0
Now, we have a quadratic equation which we can solve using various methods such as factoring, completing the square, or using the quadratic formula.
Alternatively, you can simply plug the equation into an online quadratic equation solver or use software like Microsoft Excel or Google Sheets to find the numerical solution for "x".
Once you find the value of "x", you can substitute it back into the expression for the other side (2x - 10) to find its length.