Final answer:
Using the Pythagorean theorem, we find that the hypotenuse lengths for the first and second triangles are 5 and 50 units, respectively. The diagonal of a square with side length 7 inches is approximately 9.9 inches. The distance between the two points (0, -3) and (3, 4) is about 7.6 units, and the perimeter of the given triangle is 15 units.
Step-by-step explanation:
The length of the hypotenuse in a right-angled triangle can be found using the Pythagorean theorem, which states that for a right-angled triangle with legs of lengths a and b, and hypotenuse of length c, the following equation holds true: a² + b² = c².
For the first triangle with leg lengths 3 and 4, the hypotenuse can be calculated as follows: √(3² + 4²) = √(9 + 16) = √25 = 5. For the second triangle with leg lengths 14 and 48, the calculation is: √(14² + 48²) = √(196 + 2304) = √2500 = 50.
The length of a square's diagonal is also calculated using the Pythagorean theorem, as the square's diagonal and sides form two right-angled triangles. For a square with side lengths of 7 inches, the diagonal length is: √(7² + 7²) = √(49 + 49) = √98 ≈ 9.9 inches when rounded to the nearest tenth.
To find the distance between two points in a plane, the formula √((x2 - x1)² + (y2 - y1)²) is used. The distance between (0, -3) and (3, 4) is: √((3 - 0)² + (4 - (-3))²) = √(9 + 49) = √58 ≈ 7.6 when rounded to the nearest tenth.
The perimeter of a triangle can be determined by finding the lengths of its sides and then summing them. For a triangle with vertices (2,4), (5,4), and (2, -2), the side lengths are calculated as follows: the horizontal side on the top has a length of 5 - 2 = 3 units, the vertical side on the right has a length of 4 - (-2) = 6 units, and the vertical side on the left is straight, sharing the same x-coordinate, with a length of 4 - (-2) = 6 units. The perimeter is thus 3 + 6 + 6 = 15 units.