Question

1. given the rectangle abcd. ad = 9, ac = 22, and m∠bca = 66°. determine the missing measures. bc = m∠adc = ab = m∠bac = bd = m∠cdb = ec = m∠aeb =

Explanation:

Step1: Properties of rectangle

In a rectangle \(ABCD\), \(AD = BC\). Given \(AD = 9\), so \(BC=9\).

Step2: Right - angle property

Since \(ABCD\) is a rectangle, \(\angle ADC = 90^{\circ}\).

Step3: Using trigonometry in \(\triangle ABC\)

\(\cos\angle BCA=\frac{BC}{AC}\), \(AC = 22\), \(BC = 9\). Also, in right - triangle \(ABC\), \(\angle BAC+\angle BCA = 90^{\circ}\). So \(\angle BAC=90^{\circ}-\angle BCA\). Given \(\angle BCA = 66^{\circ}\), then \(\angle BAC = 24^{\circ}\).

Step4: Diagonals of rectangle

In a rectangle, diagonals are equal. So \(BD = AC=22\).

Step5: Angles in rectangle and triangle

\(\angle CDB=\angle BAC\) (alternate interior angles for \(AB\parallel CD\) and diagonals as transversal). So \(\angle CDB = 24^{\circ}\).

Step6: Mid - point of diagonals

Since diagonals of a rectangle bisect each other, \(EC=\frac{AC}{2}\). Given \(AC = 22\), so \(EC = 11\).

Step7: Angle at intersection of diagonals

\(\angle AEB = 2\angle BCA\) (using the property that the angle between the diagonals and the angle at the side of the rectangle. \(\angle AEB=132^{\circ}\) (because \(\angle BCA = 66^{\circ}\) and \(\angle AEB\) is the exterior angle of \(\triangle BEC\) where \(BE = EC\))

Answer:

\(BC = 9\), \(\angle ADC=90^{\circ}\), \(\angle BAC = 24^{\circ}\), \(BD = 22\), \(\angle CDB=24^{\circ}\), \(EC = 11\), \(\angle AEB=132^{\circ}\)