Question

1. is the triangle shown a right triangle? explain your reasoning. image: three squares (areas 50 cm², 40 cm², 20 cm²) attached to the sides of a blue triangle

Explanation:

Step1: Recall Pythagorean theorem

For a right triangle, the sum of the squares of the two shorter sides (legs) equals the square of the longest side (hypotenuse). Let the areas of the squares on the legs be \( A_1, A_2 \) and on the hypotenuse be \( A_3 \). Then \( A_1 + A_2 = A_3 \) if it's a right triangle.

Step2: Identify areas of squares

Here, the areas of the squares are \( 20 \, \text{cm}^2 \), \( 40 \, \text{cm}^2 \), and \( 50 \, \text{cm}^2 \). Let's check if the sum of the two smaller areas equals the largest area.
Sum of smaller areas: \( 20 + 40 = 60 \)
Largest area: \( 50 \)
Since \( 20 + 40
eq 50 \), the triangle does not satisfy the Pythagorean theorem.

Answer:

No, the triangle is not a right triangle. Because for a right triangle, the sum of the areas of the squares on the two legs should equal the area of the square on the hypotenuse (by the Pythagorean theorem). Here, \( 20 + 40 = 60 \), which is not equal to \( 50 \), so it does not satisfy the Pythagorean theorem.