Find the sides of the rectangle if you know that one of them is 14 cm larger than the other, and the diagonal of the rectangle is 34 cm.
Since the diagonal of the rectangle divides it into two equal right-angled triangles, the sides of the rectangle will be the legs of this triangle, and the diagonal will be its hypotenuse.
Using the Pythagorean theorem, we compose an equation, where x is one of the sides of the rectangle, and x + 14 is its other side:
x ^ 2 + (x + 14) ^ 2 = 34 ^ 2;
x ^ 2 + x ^ 2 + 28x + 196 = 1156;
2x ^ 2 + 28x – 960 = 0.
Reduce the equation by 2:
x ^ 2 + 14x – 480 = 0.
The root of the discriminant is 46.
x1 = (-14 – 46) / 2 = -30 is a negative number, therefore it is not suitable for solving the problem;
x2 = (-14 + 46) / 2 = 16 cm – the length of one of the sides of the rectangle;
16 + 14 = 30 cm – the length of the other side of the rectangle.
Answer: 16 cm, 30 cm.
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