a According to the Triangle Inequality Theorem, the sum of the lengths of any two sides of a triangle is greater than the length of the third side. Let's try this for the triangle's sides.
8 + 15? >17& ⇔ 23>17
8 + 17? >15& ⇔ 25>15
15 + 17? >8& ⇔ 32>8Therefore, a triangle with the given measurements exists. If this is a right triangle, we are able to substitute the triangles sides in the Pythagorean Theorem and the equation will hold true.
8 + 12&? >4 ⇔ 20>4
12 + 4&? >8 ⇔ 16>8
8 + 4&? >12 ⇔ 12≯12
Since the sum of the two smaller sides isn't greater than the longest side, we cannot create a triangle with the given side lengths.
