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A right triangle with side lengths R, S, and T is formed with three squares joined at their vertices as shown below.

The image illustrates the Pythagorean theorem with three squares attached to the sides of a right triangle. The squares have areas of 324 cm² and 900 cm², representing the squares of the lengths of the two shorter sides of the right triangle. The shaded triangle inside is a right triangle, with "Side R", "Side S", and "Side T" labeling its sides. The square attached to the hypotenuse (Side T) has an area of 900 cm², and the square attached to one of the legs (Side S) has an area of 324 cm². According to the Pythagorean theorem, the sum of the areas of the squares on the legs (a² + b²) equals the area of the square on the hypotenuse (c²). Therefore, the area of the square on the remaining leg (Side R) can be calculated by subtracting the area of the square on Side S from the area of the square on Side T (900 cm² - 324 cm² = 576 cm²). The length of each side can be found by taking the square root of the respective square's area.

Using the areas of the given squares, what are the side lengths of the R, S, & T?

The length of Side R is
.

The length of Side S is
.

The length of Side T is
.

1 answer

To find the side lengths of the triangle, we start with the areas of the squares.

  1. Area of the square attached to Side T (hypotenuse):
    Given area = 900 cm²
    Therefore, Side T = √(900) = 30 cm.

  2. Area of the square attached to Side S (one of the legs):
    Given area = 324 cm²
    Therefore, Side S = √(324) = 18 cm.

  3. Area of the square attached to Side R (the remaining leg):
    We know from the Pythagorean theorem that a² + b² = c².
    We know the following:

    • Side S² = 324 cm² (this corresponds to Side S, one leg)
    • Side T² = 900 cm² (this corresponds to Side T, the hypotenuse)
    • We need to find Side R.

    Using the formula:
    Side R² = Side T² - Side S²
    Side R² = 900 cm² - 324 cm²
    Side R² = 576 cm²
    Therefore, Side R = √(576) = 24 cm.

Now we can summarize:

  • The length of Side R is 24 cm.
  • The length of Side S is 18 cm.
  • The length of Side T is 30 cm.