Final answer:
To find the length of the hypotenuse in a right-angled triangle given the lengths of the other two sides and the area, use the area formula to find the side lengths, then the Pythagorean theorem to calculate the hypotenuse.
Step-by-step explanation:
The question involves finding the length of the hypotenuse of a right-angled triangle when given the lengths of the other two sides and the area. The sides are 4x cm and x+1 cm, and the area of the triangle is 84 cm². By the formula for the area of a right-angled triangle (1/2 × base × height), we can set up an equation: 1/2 × (4x) × (x+1) = 84. Solving for x gives us the lengths of the two legs of the triangle. Once we have the values of x, we can calculate the length of the hypotenuse using the Pythagorean theorem, which is a² + b² = c² (where a and b are the legs of the triangle and c is the hypotenuse). The hypotenuse c can be found by taking the square root of the sum of the squares of the other two sides.