joey is building a frame for a sandbox. the sandbox is going to be a quadrilateral that has the lengths…

Explanation:

Step1: Recall rectangle properties

A rectangle has opposite - sides equal and all angles are right - angles. In a right - triangle formed by the sides and diagonal of a rectangle, the Pythagorean theorem (a^{2}+b^{2}=c^{2}) holds, where (c) is the diagonal and (a) and (b) are the sides of the right - triangle.

Step2: Check if Pythagorean theorem holds

Let's consider the right - triangle with sides (a = 8) ft and (b = 12) ft. Calculate (a^{2}+b^{2}): [a^{2}+b^{2}=8^{2}+12^{2}=64 + 144=208] The square of the diagonal (c = 14) ft, so (c^{2}=14^{2}=196). Since (a^{2}+b^{2}\neq c^{2}), the angles are not right - angles.

Step3: Analyze the shape

The sandbox is a quadrilateral because it has four sides. We know that it is not a rectangle because the Pythagorean theorem does not hold for the right - triangles formed by its sides and diagonal. Also, we cannot determine the nature of angles (C) and (X) (acute or obtuse) from the given information about side lengths and diagonal length alone, but we know it is a quadrilateral.

Answer:

a quadrilateral, because angle C and angle X are acute (Note: The reason for choosing this option among the given ones is that we know it is a quadrilateral and we have no information to suggest right - angles or obtuse angles, and in non - rectangular quadrilaterals with the given side lengths, the non - right angles are likely to be acute, but the main point is that it is a quadrilateral)