The Pythagorean theorem can be applied to which of the following triangles? Triangle A: A triangle with sides 1, 1 and square root of 2Triangle B: An equilateral triangleTriangle C: A scalene triangle, with length 7, 8 and 9Triangle D: A triangle that is inscribed in a semicircle, where the diameter of the semicircle is its longest side

The Pythagorean theorem can be applied to right triangles. It states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This can be written as: a² + b² = c².

Let's examine each triangle:

Triangle A: This is a right triangle (also known as an isosceles right triangle or a 45-45-90 triangle). The sides follow the pattern 1:1:√2, which fits the Pythagorean theorem (1² + 1² = (√2)²). So, the Pythagorean theorem can be applied to Triangle A.

Triangle B: An equilateral triangle is not a right triangle, as all its angles are 60 degrees. Therefore, the Pythagorean theorem cannot be applied to Triangle B.

Triangle C: This is a scalene triangle (a triangle that has all sides of different lengths). However, not all scalene triangles are right triangles. To check if it's a right triangle, we can use the Pythagorean theorem. For Triangle C, 7² + 8² ≠ 9², so it's not a right triangle. Therefore, the Pythagorean theorem cannot be applied to Triangle C.

Triangle D: A triangle inscribed in a semicircle with the diameter as its longest side is always a right triangle (this is known as the Inscribed Angle Theorem or Thales' Theorem). Therefore, the Pythagorean theorem can be applied to Triangle D.

In conclusion, the Pythagorean theorem can be applied to Triangles A and D.