Question

1. mp.3 construct arguments which triangles are right triangles? construct a math argument to justify your answer.

Explanation:

To determine which triangles are right triangles, we use the Pythagorean theorem: A triangle with side lengths \( a \), \( b \), and hypotenuse \( c \) (where \( c \) is the longest side) is a right triangle if and only if \( a^2 + b^2 = c^2 \).

Step 1: Identify Triangle Sides

From the diagram, the yellow triangle has a leg labeled \( 4 \), and we assume the other leg is equal to the side of the square (let’s say the square has side length \( s \), so the other leg is \( s \)), and the hypotenuse is the diagonal of the square’s corner triangle. Wait, actually, if the triangle is part of a rectangle or square, the right angle is implied by the corner. But to confirm, let’s check the Pythagorean theorem.

For the yellow triangle:

• If it’s a right triangle, one angle should be \( 90^\circ \). Since it’s drawn with a right angle (the corner of the square/rectangle), by definition, a triangle with a \( 90^\circ \) angle is a right triangle.

For other triangles (e.g., the orange, red, blue ones):

• If they share the right angle corner or have sides that satisfy \( a^2 + b^2 = c^2 \), they are right triangles. But based on the diagram, the yellow triangle (and likely others with a right angle at the square’s corner) is a right triangle because it has a right angle (the corner of the square/rectangle forms a \( 90^\circ \) angle).

Final Argument

A right triangle has one \( 90^\circ \) (right) angle. The yellow triangle (and any triangle with a right angle at the square’s corner) is a right triangle because:

1. The corner of the square/rectangle is a \( 90^\circ \) angle.
2. A triangle with a \( 90^\circ \) angle is a right triangle (by definition of a right triangle).

Answer:

The yellow triangle (and other triangles with a right angle at the square’s corner) is a right triangle because it contains a \( 90^\circ \) angle (from the square’s corner), satisfying the definition of a right triangle.