Question

1. choose the correct answer.

image of a triangle with sides √12.96 cm, 3 cm, 4 cm

a = 3 cm, b = √12.96 cm, c = 4 cm
the triangle is a(n) ____ triangle.

options:

• obtuse
• right
• acute

Explanation:

Step1: Recall the Pythagorean theorem

For a triangle with sides \(a\), \(b\), \(c\) (where \(c\) is the longest side), if \(a^{2}+b^{2}=c^{2}\), it is a right triangle; if \(a^{2}+b^{2}c^{2}\), it is an acute triangle. First, find the longest side. Let's calculate the numerical values: \(a = 3\), \(b=\sqrt{12.96}=3.6\), \(c = 4\). So the longest side is \(c = 4\).

Step2: Calculate \(a^{2}+b^{2}\) and \(c^{2}\)

Calculate \(a^{2}+b^{2}\): \(a^{2}=3^{2} = 9\), \(b^{2}=(\sqrt{12.96})^{2}=12.96\), so \(a^{2}+b^{2}=9 + 12.96=21.96\). Calculate \(c^{2}=4^{2}=16\).

Step3: Compare \(a^{2}+b^{2}\) and \(c^{2}\)

Since \(21.96>16\) (i.e., \(a^{2}+b^{2}>c^{2}\)), the triangle is acute.

Answer:

acute (the option corresponding to "acute")