Question

what is the area of an isosceles trapezoid if the lengths of its bases are 16 cm and 30 cm and the diagonals are perpendicular to each other? answer: (a=square cm^{2})

Explanation:

Step1: Translate the trapezoid

By translating one of the diagonals of the isosceles - trapezoid, we can get a right - angled isosceles triangle. Let \(ABCD\) be an isosceles trapezoid with \(AD = 16\mathrm{cm}\), \(BC = 30\mathrm{cm}\), and \(AC\perp BD\), and \(AC = BD\). Translate \(AC\) parallel to \(AD\) and extend \(BC\) to \(E\). Since \(AD\parallel BC\) and \(AC\parallel DE\), the quadrilateral \(ACED\) is a parallelogram. Then \(AD = CE=16\mathrm{cm}\), \(AC = DE\), and \(AC\parallel DE\). Because \(AC\perp BD\), so \(DE\perp BD\). And since the trapezoid \(ABCD\) is isosceles, \(AC = BD\), then \(BD = DE\).

Step2: Calculate the base of the right - angled isosceles triangle

The length of \(BE\) is \(BE=BC + CE=30 + 16=46\mathrm{cm}\).

Step3: Find the area of the right - angled isosceles triangle

The area of a right - angled triangle \(S=\frac{1}{2}\times BD\times DE\). In right - angled isosceles triangle \(BDE\) (\(BD = DE\)), and the area of the trapezoid \(S_{trapezoid}=S_{\triangle BDE}\) (because \(S_{trapezoid}=S_{\triangle ABD}+S_{\triangle BCD}\), \(S_{\triangle ABD}=S_{\triangle ACD}\), and \(S_{\triangle ACD}=S_{\triangle CDE}\)). For right - angled isosceles triangle \(BDE\) with right - angle sides \(BD\) and \(DE\), and \(BE\) as the hypotenuse. According to the property of right - angled isosceles triangle, if the length of the hypotenuse is \(l\), and the length of the right - angle side is \(a\), then \(l=\sqrt{2}a\), and the area \(S=\frac{1}{2}a^{2}\). Also, since \(S=\frac{1}{2}\times BD\times DE\) and \(BD = DE\), and \(BE = BD + DE\) (in right - angled isosceles triangle \(BDE\)), the area of \(\triangle BDE\) is \(S=\frac{1}{2}\times BD\times DE=\frac{1}{2}\times\frac{BE^{2}}{2}\). Substitute \(BE = 46\mathrm{cm}\) into the formula, we get \(S=\frac{1}{2}\times\frac{46^{2}}{2}=\frac{46\times46}{4}=529\mathrm{cm}^{2}\).

Answer:

\(529\)