Question

1. choose the correct answer.

a pythagorean triple is a a set of three ____ positive whole numbers, a, b, and c, such that $a^2 + b^2 = c^2$.

• nonzero
• irrational

1. choose the correct answer.

image of a triangle with a = 13 mm, b = 84 mm, c = 85 mm
the triangle is a(n) ____ triangle.

• right
• obtuse
• acute

Explanation:

Question 1

Step1: Recall Pythagorean Triple definition

A Pythagorean Triple consists of three positive whole numbers (integers) that satisfy \(a^2 + b^2 = c^2\). These numbers must be nonzero because if any were zero, the equation wouldn't represent a triangle's side lengths (a triangle can't have a side length of 0). "Irrational" is incorrect because Pythagorean Triples use whole (rational) numbers.

Step2: Eliminate incorrect option

The option "irrational" is wrong as Pythagorean Triples are made of whole numbers (rational). So the correct term is "nonzero".

Step1: Check Pythagorean theorem

For a triangle with sides \(a = 13\), \(b = 84\), \(c = 85\), we check if \(a^2 + b^2 = c^2\). Calculate \(a^2 = 13^2 = 169\), \(b^2 = 84^2 = 7056\), sum them: \(169 + 7056 = 7225\). Now \(c^2 = 85^2 = 7225\).

Step2: Determine triangle type

Since \(a^2 + b^2 = c^2\), by the converse of the Pythagorean theorem, the triangle is a right triangle. Obtuse triangles have \(a^2 + b^2 < c^2\) (for \(c\) the longest side), and acute have \(a^2 + b^2 > c^2\) (for \(c\) the longest side). Here, the equation holds, so it's right.

Answer:

nonzero (the option labeled "nonzero")

Question 2