Final answer:
Using Apollonius's theorem, the length of the hypotenuse (AC), given medians AM and BN of lengths 19 and 18 respectively, is calculated to be 38. The calculation involves assuming the length of the hypotenuse to be twice the length of the given medians and then applying the theorem to solve for the missing side before finding the hypotenuse.
Step-by-step explanation:
To find the length of the hypotenuse in a right triangle when given the medians to the hypotenuse (AM and BN), we can make use of the Apollonius's theorem. This theorem states that in any triangle, the sum of the squares of any two medians equals ⅓ (2x the square of the third side plus 2x the square of the fourth side). In a right triangle, the medians to the hypotenuse are equal to half the hypotenuse. Therefore, AM and BN are each half the length of the hypotenuse (AC and BC, respectively).
So, if AM = 19, then AC, the hypotenuse, is twice that, so AC = 2 × 19 = 38. Similarly, if BN = 18, then BC is twice that, so BC = 2 × 18 = 36. Now we can use this information in the Apollonius's theorem to solve for the length of the hypotenuse.
According to the Apollonius's theorem for a right triangle: AM² + BN² = ⅓(AB² + AC²)
Let's set AB as x, then we have:
19² + 18² = ⅓(x² + 38²)
Simplifying:
361 + 324 = ⅓(x² + 1444)
685 = ⅓(x² + 1444)
×3 both sides to get rid of the fraction:
2055 = 2x² + 2888
Subtracting 2888 from both sides:
2055 - 2888 = 2x²
−1633 = 2x²
Dividing by 2:
−816.5 = x²
Taking the square root of both sides:
x = −29
So, the length of side AB is 29, and since we had obtained AC as 38, 38 is the length of the hypotenuse of the triangle.