Question

pythagorean theorem converse assignment\
possible points: 16\
given the side lengths below, first determine whether they form a triangle. if they do, determine whether the triangle is acute, right, or obtuse.\
20, 23, 41\
\bigcirc acute triangle\
\bigcirc right triangle\
\bigcirc obtuse triangle\
\bigcirc not a triangle

Explanation:

Step1: Check Triangle Inequality

To form a triangle, the sum of any two sides must be greater than the third side.

• \(20 + 23 = 43>41\)
• \(20 + 41 = 61>23\)
• \(23 + 41 = 64>20\)

So, it is a triangle.

Step2: Use Pythagorean Theorem Converse

Let \(a = 20\), \(b = 23\), \(c = 41\) (where \(c\) is the longest side).
Calculate \(a^{2}+b^{2}\) and \(c^{2}\):

• \(a^{2}=20^{2} = 400\)
• \(b^{2}=23^{2}=529\)
• \(a^{2}+b^{2}=400 + 529=929\)
• \(c^{2}=41^{2}=1681\)

Since \(a^{2}+b^{2}=929<1681 = c^{2}\), by the converse of the Pythagorean theorem (and the rule for obtuse triangles: if \(a^{2}+b^{2}

Answer:

Obtuse Triangle