Question

1. given the rectangle abcd, ad = 9, ac = 22, and ( mangle bca = 66^{circ} ). determine the missing measures.

( bc=)( mangle adc=)( ab=)( mangle bac=)( bd=)( mangle cdb=)( ec=)( mangle aeb=)

Explanation:

Step1: Properties of rectangle

In a rectangle, opposite sides are equal, so \(BC = AD = 9\).

Step2: Using Pythagorean theorem

In right - triangle \(ABC\), \(AB=\sqrt{AC^{2}-BC^{2}}\). Substitute \(AC = 22\) and \(BC = 9\), we get \(AB=\sqrt{22^{2}-9^{2}}=\sqrt{484 - 81}=\sqrt{403}\approx20.07\).

Step3: Diagonals of rectangle

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

Step4: Mid - point of diagonals

Since diagonals of a rectangle bisect each other, \(EC=\frac{1}{2}AC\). Substitute \(AC = 22\), we get \(EC = 11\).

Step5: Angle of rectangle

Each angle of a rectangle is \(90^{\circ}\), so \(m\angle ADC=90^{\circ}\).

Step6: Alternate interior angles

Since \(AD\parallel BC\), \(m\angle DAC=m\angle BCA = 66^{\circ}\). In right - triangle \(ADC\), \(m\angle BAC=90^{\circ}-m\angle DAC=90^{\circ}-66^{\circ}=24^{\circ}\).

Step7: Alternate interior angles

Since \(AB\parallel DC\), \(m\angle CDB=m\angle BAC = 24^{\circ}\).

Step8: Vertical angles

\(\angle AEB\) and \(\angle DEC\) are vertical angles. Also, \(\triangle DEC\) is isosceles (\(DE = EC\)), \(m\angle DEC=180^{\circ}-2\times24^{\circ}=132^{\circ}\), so \(m\angle AEB = 132^{\circ}\).

Answer:

\(BC = 9\), \(AB\approx20.07\), \(BD = 22\), \(EC = 11\), \(m\angle ADC=90^{\circ}\), \(m\angle BAC = 24^{\circ}\), \(m\angle CDB=24^{\circ}\), \(m\angle AEB = 132^{\circ}\)