To solve this problem, we need to use the Pythagorean theorem to find the length of the ramp, which will be the hypotenuse of a right-angled triangle. The height of the porch forms one leg of the triangle (the vertical leg), and the horizontal distance from the porch where the ramp starts forms the other leg.
Let's denote the following:
- Let \( a \) be the vertical leg of the triangle, which is the height of the porch (40 centimeters).
- Let \( b \) be the horizontal leg of the triangle, which is the distance from the porch where the ramp starts (75 centimeters).
- Let \( c \) be the hypotenuse of the triangle, which will be the length of the ramp we want to find.
The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (\( c \)) is equal to the sum of the squares of the lengths of the other two sides (\( a \) and \( b \)):
\[ a^2 + b^2 = c^2 \]
We know that \( a = 40 \) cm and \( b = 75 \) cm. Plugging these values into the theorem:
\[ 40^2 + 75^2 = c^2 \]
\[ 1600 + 5625 = c^2 \]
\[ 7225 = c^2 \]
Now we take the square root of both sides to solve for \( c \):
\[ c = \sqrt{7225} \]
\[ c = 85 \]
Therefore, the length of the ramp that Tisha needs to build is 85 centimeters.
