Question

can the sides of a triangle have lengths of 28, 24, and 10? if so, what kind of triangle is it? yes, acute yes, right yes, obtuse no

Explanation:

Step1: Check triangle - inequality theorem

The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
$10 + 24=34>28$, $10 + 28 = 38>24$, $24+28 = 52>10$. So, a triangle can be formed.

Step2: Determine the type of triangle

Let \(a = 10\), \(b = 24\), \(c = 28\). Use the Pythagorean - like relationship \(c^{2}\) compared to \(a^{2}+b^{2}\).
Calculate \(a^{2}+b^{2}\): \(a^{2}=10^{2}=100\), \(b^{2}=24^{2}=576\), so \(a^{2}+b^{2}=100 + 576=676\).
Calculate \(c^{2}\): \(c^{2}=28^{2}=784\).
Since \(c^{2}>a^{2}+b^{2}\) (i.e., \(784>676\)), the triangle is obtuse.

Answer:

yes, obtuse