Question

applications

1. a truss is a collection of beams connected at their ends that occur in many structures, such as roofs and bridges. a simple truss is shown below. it can be described geometrically as follows. point b is the image of point a after a ( 58^{circ} ) clockwise rotation about point d. point e is the image of point d after a ( 58^{circ} ) counterclockwise rotation about point b. finally, point c is the image of point b after a ( 58^{circ} ) clockwise rotation about point e.

(a) on the diagram, label all angles that have a ( 58^{circ} ) measure. state the angle names below that have these measures.
(b) what other line segments must have the same length as ( overline{bd} )?

Explanation:

Step1: Recall rotation properties

Rotation preserves angle measures and segment lengths.

Step2: Identify angles with \(58^{\circ}\) measure

Since \(B\) is the image of \(A\) after a \(58^{\circ}\) clock - wise rotation about \(D\), \(\angle ADB = 58^{\circ}\). Since \(E\) is the image of \(D\) after a \(58^{\circ}\) counter - clockwise rotation about \(B\), \(\angle DBE=58^{\circ}\). Since \(C\) is the image of \(B\) after a \(58^{\circ}\) clock - wise rotation about \(E\), \(\angle BEC = 58^{\circ}\).

Step3: Identify segments equal to \(BD\)

By rotation, \(AD = BD\) (rotation about \(D\) maps \(A\) to \(B\)), \(BE = BD\) (rotation about \(B\) maps \(D\) to \(E\)), \(CE=BE\) (rotation about \(E\) maps \(B\) to \(C\)). So \(AD = BE=CE\)

Answer:

(a) The angles with \(58^{\circ}\) measure are \(\angle ADB\), \(\angle DBE\), \(\angle BEC\).
(b) The line segments \(AD\), \(BE\), \(CE\) have the same length as \(BD\).