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: Analyze the properties of a square

In a square, the diagonals are equal and bisect each other at right - angles. Also, using the Pythagorean theorem for a right - triangle formed by half of the diagonals and a side of the square. Let the length of the diagonal be \(d\) and the side be \(s\). If the diagonal \(d = 48\) inches, then half of the diagonal \(a=\frac{d}{2}=24\) inches.

Step2: Apply the Pythagorean theorem

According to the Pythagorean theorem \(a^{2}+b^{2}=s^{2}\). In a square, if we assume the other diagonal is also \(d\) (but here we just check for the right - angle property using the given diagonal and side). We know \(s = 25\) inches and \(a = 24\) inches. Let's check if \(a^{2}+b^{2}=s^{2}\). If we assume the other half - diagonal \(b\) (in case of a square \(a = b\)), but actually, for a quadrilateral with side \(s = 25\) and half - diagonal \(a=24\), if it were a right - triangle (formed by half - diagonal and side), then by Pythagorean theorem \(24^{2}+b^{2}=25^{2}\). We know that \(25^{2}-24^{2}=(25 + 24)(25 - 24)=49\), so \(b = 7\) (not equal to \(24\) if it were a square. Also, in a square, if the side \(s\) and diagonal \(d\) are related as \(d=\sqrt{2}s\). If \(s = 25\), then \(d=\sqrt{2}\times25\approx35.35
eq48\). Another way: If we consider the triangle formed by half of the diagonal (\(24\) inches) and the side (\(25\) inches). If the angle at the vertex (where the side and half - diagonal meet) were a right - angle, then by Pythagorean theorem \(24^{2}+24^{2}\) (for a square, both half - diagonals are equal) should equal \(25^{2}\). But \(24^{2}+24^{2}=576+576 = 1152\) and \(25^{2}=625\). Also, using the property that in a square, if the side is \(s\) and diagonal \(d\), \(d^{2}=2s^{2}\). Here \(d = 48\), \(d^{2}=2304\) and \(2s^{2}=2\times625 = 1250
eq2304\). The key is that in a square, the sides and diagonals satisfy \(d=\sqrt{2}s\) (derived from Pythagorean theorem for the right - angled triangle formed by two adjacent sides and a diagonal). Since \(48
eq\sqrt{2}\times25\approx35.35\), 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.