The difference between the sides of a right-angled triangle is 9 cm and the area is 486 cm. Find the perimeter and area.

According to the condition of the problem, let the sides of a right-angled triangle be a and b, respectively. If the difference between the sides = 9, and the area of ​​a right-angled triangle is found by the formula 1/2 * a * b, then we get a system of equations from two unknowns. Then:

a – b = 9,

1/2 * a * b = 486,

Let us express one unknown from one equation and substitute the resulting expression into the second equation:

a = 9 + b,

b * (9 + b) = 486 * 2,

We will leave the first equation unchanged for now, and in the second equation we will open the brackets and give similar terms:

9b + b ^ 2 = 972,

We got a quadratic equation, which needs to be reduced to a standard form:

b ^ 2 + 9b – 972 = 0,

We solve the quadratic equation through the discriminant:

b = (-9 +/- √ (81 – 4 * 1 * (-972))) / 2,

The radical expression is √3888, which is approximately = 62.

Now we find the roots of the equation:

b1 = (-9 + 62) / 2 = 26.5;

b2 = (-9 – 62) / 2 = -35.5.

Since b is the side of the triangle, it cannot be negative, i.e. only one value remains: b = 26.5.

we substitute this obtained value into the first equation of the system:

a = b + 9 = 26.5 + 9 = 35.5.

Check: the difference between the sides of the triangle is 35.5 – 26.5 = 9, and the area of ​​a right-angled triangle is 1/2 * 35.5 * 26.5 = 470 (which is approximately equal to the specified area value 486 – the error appeared as a result of extracting the square root ).

The area of ​​a triangle is the sum of all sides. The unknown side is the hypotenuse, which we find by the Pythagorean theorem:

√ (35.5 ^ 2 + 26.5 ^ 2) = √ (1260.25 + 702.25) = √1962.5 = 44.

The perimeter of the triangle is:

35.5 + 26.5 + 44 = 106.