(please explain with brief explanation)
The sides of an equilateral triangle are shortened by 12 units,13 units and 14 units respectively and a right angled triangle is formed.Find the side of the equilateral triangle.
The side shortened least is longest, so
(x-14)^2 + (x-13)^2 = (x-12)^2
Now, we all know that the 3-4-5 triangle is the only one with sides which are consecutive integers. So, x-14=3, making x=17.
Solving the equation gives the same result.
To find the side of the equilateral triangle, we can use the Pythagorean theorem because we have a right-angled triangle.
The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
In this case, the sides of the right-angled triangle are shortened sides of the equilateral triangle. Let's assume the original side length of the equilateral triangle is "x".
According to the problem, the sides are shortened by 12 units, 13 units, and 14 units. So the lengths of the sides of the right-angled triangle are (x-12), (x-13), and (x-14).
Now, let's apply the Pythagorean theorem:
(x-12)^2 + (x-13)^2 = (x-14)^2
Expanding and simplifying the equation, we get:
x^2 - 24x + 144 + x^2 - 26x + 169 = x^2 - 28x + 196
Combining like terms and canceling out the x^2 term on both sides, we get:
x^2 - 2x - 137 = 0
We can solve this quadratic equation using factoring, completing the square, or by using the quadratic formula. Once we solve for x, we will have the side length of the equilateral triangle.