Question

the diagonal of rectangle abcd measures 2 inches in length
what is the length of line segment ab?
4√3 inches
1 inch
4 inches
√3 inches

Explanation:

Step1: Identify the triangle type

In rectangle \(ABCD\), \(\triangle ABD\) is a right - triangle (since \(ABCD\) is a rectangle, \(\angle A = 90^{\circ}\)). Given \(\angle ADB=30^{\circ}\) and hypotenuse \(BD = 2\) inches.

Step2: Use the sine function

In a right - triangle, \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). For \(\theta = 30^{\circ}\) in \(\triangle ABD\), if \(\theta=\angle ADB\), and the side opposite to \(\angle ADB\) is \(AB\).
We know that \(\sin30^{\circ}=\frac{AB}{BD}\).
Since \(\sin30^{\circ}=\frac{1}{2}\) and \(BD = 2\) inches.
Substitute the values into the formula: \(\frac{1}{2}=\frac{AB}{2}\).

Step3: Solve for \(AB\)

Cross - multiply: \(AB=\sin30^{\circ}\times BD\).
\(AB=\frac{1}{2}\times2 = 1\) inch.

Answer:

\(1\) inch