Question

in trapezoid abcd with legs \\( \overline{ab} \\) and \\( \overline{cd} \\), point o is the intersection of the diagonals. if \\( a_{abo}=6 \text{ in.}^2 \\), find \\( a_{cod} \\). answer: \\( a_{doc}= \\) in.²

Explanation:

Step1: Use the property of trapezoid diagonals

In a trapezoid \(ABCD\) with \(AB\parallel CD\), \(\triangle ABO\) and \(\triangle CDO\) are similar. Also, \(S_{\triangle ABC}=S_{\triangle ABD}\) (because they have the same base \(AB\) and the same height between the parallel lines \(AB\) and \(CD\)). Then \(S_{\triangle ABC}-S_{\triangle ABO}=S_{\triangle ABD}-S_{\triangle ABO}\).

Step2: Calculate the area of \(\triangle COD\)

Since \(S_{\triangle ABC}-S_{\triangle ABO}=S_{\triangle ACO}\) and \(S_{\triangle ABD}-S_{\triangle ABO}=S_{\triangle BDO}\), and \(S_{\triangle ACO} = S_{\triangle BDO}\). Also, \(S_{\triangle ADO}+S_{\triangle ACO}=S_{\triangle BCO}+S_{\triangle BDO}\). We know that \(S_{\triangle ABO}\) and \(S_{\triangle CDO}\) satisfy the property: In trapezoid \(ABCD\) with \(AB\parallel CD\), \(S_{\triangle ABO}=S_{\triangle CDO}\) (because \(\triangle ABC\) and \(\triangle ABD\) have equal area, subtract the common part \(\triangle ABO\) and use the ratio of similar - triangles and the relationship between areas of triangles with the same height or base). Given \(A_{ABO} = 6\space in^{2}\), so \(A_{COD}=6\space in^{2}\).

Answer:

\(6\)