Question

claire is designing a banner that will hang in her classroom. the length of one diagonal of the banner is 48 inches, and the sides are 25 inches long. is the banner a square? it is not a square because opposite angles are not congruent. it is not a square because the angles at the vertices are not right angles. it is a square because all the sides are congruent. it is a square because the diagonals are perpendicular.

Explanation:

Step1: Recall properties of a square

In a square, if we consider half - diagonals and a side, they form a right - triangle due to the diagonals of a square being perpendicular bisectors of each other. Let the diagonal be $d = 48$ inches, so half - diagonal $a=\frac{d}{2}=24$ inches and side $s = 25$ inches.

Step2: Apply Pythagorean theorem

In a right - triangle, if the sides of the right - triangle are $a$, $b$ (half - diagonals) and $s$ (side of the quadrilateral), by Pythagorean theorem $s^{2}=a^{2}+b^{2}$. Here, if it were a square with perpendicular diagonals, and $a = 24$ inches, $b = 24$ inches, then $a^{2}+b^{2}=24^{2}+24^{2}=576 + 576=1152$, and $s^{2}=25^{2}=625$. Since $a^{2}+b^{2}
eq s^{2}$, the angles at the vertices are not right angles.

Answer:

It is not a square because the angles at the vertices are not right angles.