Question

pythagorean theorem converse assignment
given the side lengths below, first determine whether they form a triangle. if they do, determine whether the triangle is acute, right, or obtuse
15, 16, 21
acute triangle
right triangle
obtuse triangle
not a triangle

Explanation:

Step1: Check triangle inequality

First, check if the three sides \(15\), \(16\), and \(21\) can form a triangle. The triangle inequality states that the sum of any two sides must be greater than the third side.

• \(15 + 16 = 31>21\)
• \(15 + 21 = 36>16\)
• \(16 + 21 = 37>15\)

So, they can form a triangle.

Step2: Use the converse of Pythagorean theorem and triangle type rule

Let \(a = 15\), \(b = 16\), \(c = 21\) (where \(c\) is the longest side). We use the formula \(a^{2}+b^{2}\) and \(c^{2}\) to determine the type of triangle.

• Calculate \(a^{2}+b^{2}\): \(15^{2}+16^{2}=225 + 256=481\)
• Calculate \(c^{2}\): \(21^{2} = 441\)

Since \(a^{2}+b^{2}=481>441 = c^{2}\), by the rule for triangle types: if \(a^{2}+b^{2}>c^{2}\) (where \(c\) is the longest side), the triangle is acute.

Answer:

Acute Triangle