Question

joey is building a frame for a sandbox. the sandbox is going to be a quadrilateral that has the lengths shown. if the diagonal of the sandbox measures 14 feet, which best describes the shape of the sandbox? a rectangle, because angle c and angle x are congruent a quadrilateral, because angle c and angle x are obtuse a quadrilateral, because angle c and angle x are acute a rectangle, because angle c is a right angle

Explanation:

Step1: Check the Pythagorean theorem

For a right - triangle, \(a^{2}+b^{2}=c^{2}\). Here, \(a = 8\), \(b = 12\), and \(c\) is the diagonal.
\(8^{2}+12^{2}=64 + 144=208
eq14^{2}=196\). But wait, we can also use the property of rectangles. In a rectangle, opposite sides are equal and all angles are right - angles.

Step2: Analyze the options

• A rectangle has four right - angles. If we consider the triangles formed by the diagonal of a quadrilateral with opposite sides equal (\(8\) ft and \(12\) ft). Using the Pythagorean theorem for a right - triangle with legs \(a\) and \(b\) and hypotenuse \(c\), if \(\angle C\) (or \(\angle X\)) is a right - angle, then \(a^{2}+b^{2}=c^{2}\). But we can also note that in a rectangle, by the definition (a parallelogram with four right - angles and opposite sides equal). The given quadrilateral has opposite sides equal (\(8\) ft and \(12\) ft) and if we assume the angle is right (using the fact that in a rectangle the diagonals bisect each other and the basic property of rectangle structure). The other options:
• Just being a quadrilateral (all the figures in the problem are quadrilaterals) is too general. Congruent angles (\(\angle C\) and \(\angle X\)) or angles being acute/obtuse do not guarantee the shape is a rectangle. A rectangle is a special type of quadrilateral with four right - angles.

Answer:

a rectangle, because angle \(C\) is a right angle