Please give 6 math problems based on the Pythagorean theorem, from easy to difficult, to test whether the child has fully mastered it. You can even have math competition questions.
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Let's solve our first Pythagorean theorem problem. We have a right triangle with legs of 3 and 4 centimeters. Using the formula a squared plus b squared equals c squared, we substitute: 3 squared plus 4 squared equals c squared. This gives us 9 plus 16 equals 25, so c equals 5 centimeters.
Now let's solve problem 2. We have a right triangle with a hypotenuse of 13 inches and one leg of 5 inches. We need to find the other leg. Using the Pythagorean theorem, 5 squared plus b squared equals 13 squared. This gives us 25 plus b squared equals 169. Subtracting 25 from both sides, b squared equals 144, so b equals 12 inches.
Problem 3 is a real-world application. A 10-foot ladder leans against a wall, with its base 6 feet from the wall. We need to find how high the ladder reaches. The ladder, wall, and ground form a right triangle. Using the Pythagorean theorem: 6 squared plus h squared equals 10 squared. This gives us 36 plus h squared equals 100. Solving, h squared equals 64, so h equals 8 feet.
Problem 4 involves finding the diagonal of a rectangle. We have a rectangular garden that is 12 meters long and 5 meters wide. The length, width, and diagonal form a right triangle. Using the Pythagorean theorem: 12 squared plus 5 squared equals d squared. This gives us 144 plus 25 equals 169, so d equals 13 meters.
Now let's tackle the advanced problems. Problem 5: A square with diagonal 8 root 2 centimeters. Using the Pythagorean theorem on the square's diagonal, we get s squared plus s squared equals 8 root 2 squared, giving us 2s squared equals 128, so s equals 8 centimeters and the perimeter is 32 centimeters. Problem 6: An isosceles triangle with sides 13, 13, and 10 centimeters. The height creates two right triangles. Using 5 squared plus h squared equals 13 squared, we find h equals 12 centimeters, giving an area of 60 square centimeters.