Question

quadrilateral efgh has the given measurements.

• ef = 8 units
• fg = 15 units
• gh = 8 units
• eh = 15 units
• fh = 17 units
• eg = 17 units

what conclusion can be made from the given information?
○ a. the quadrilateral is a parallelogram, but not a rectangle.
○ b. the quadrilateral is both a parallelogram and a rectangle.
○ c. the quadrilateral is neither a parallelogram nor a rectangle.
○ d. the quadrilateral is a rectangle, but not a parallelogram.

Explanation:

Step1: Check if it is a parallelogram

In a parallelogram, opposite sides are equal. Here, \(EF = 8\) units and \(GH=8\) units, \(FG = 15\) units and \(EH = 15\) units. So, by the property of a parallelogram (opposite sides are equal), quadrilateral \(EFGH\) is a parallelogram.

Step2: Check if it is a rectangle

In a rectangle, the diagonals are equal and the Pythagorean theorem holds for the sides and the diagonal. Let's check using the Pythagorean theorem. For a right - angled triangle (if the parallelogram is a rectangle, adjacent sides and diagonal form a right - triangle). Consider the sides \(EF = 8\) and \(FG = 15\). Using the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(a = 8\), \(b = 15\). Then \(a^{2}+b^{2}=8^{2}+15^{2}=64 + 225=289\), and \(c=\sqrt{289}=17\). The diagonals \(FH = 17\) and \(EG = 17\) (diagonals of a parallelogram are equal in a rectangle). Also, since \(EF^{2}+FG^{2}=FH^{2}\) (by Pythagorean theorem), the angles are right - angles.

Answer:

B. The quadrilateral is both a parallelogram and a rectangle.