To construct the quadrilateral PQRS given the conditions \(PQ = 8 , \text{cm}\), \(\angle QPS = 105^\circ\), \(\angle PQS = 30^\circ\), \(PR = 9 , \text{cm}\), and \(RS = RQ\) (which implies triangle \(RQS\) is isosceles), follow these steps:
Step 1: Draw the Line Segment PQ
- Draw a line segment \(PQ\) that measures \(8 , \text{cm}\). Label the endpoints as \(P\) and \(Q\).
Step 2: Construct Angle QPS
- At point \(P\), construct the angle \(\angle QPS = 105^\circ\). You can use a protractor to measure and mark this angle.
Step 3: Construct Angle PQS
- At point \(Q\), construct the angle \(\angle PQS = 30^\circ\). Again, use a protractor for accurate measurement.
Step 4: Locate Point S
- Draw lines along the angles made at points \(P\) and \(Q\). The lines will intersect at point \(S\).
Step 5: Measure and Draw PR
- From point \(P\), measure \(PR = 9 , \text{cm}\) along the line you constructed in Step 2. Mark this point as \(R\).
Step 6: Construct Equal Lengths RS and RQ
- Since \(RS = RQ\), point \(R\) must be such that \(RS\) and \(RQ\) are the same length. This means you'll need to use a compass:
- Set the compass to the length \(RQ\) (the distance from point \(R\) to point \(Q\)).
- Place the compass point on \(R\) and mark an arc that intersects the line that goes through point \(S\).
Step 7: Complete the Quadrilateral
- The intersection point of the arc with the line through \(S\) will give you point \(S\). This point should maintain the equal lengths \(RS\) and \(RQ\).
- Connect points \(R\) and \(S\) to complete the quadrilateral \(PQRS\).
Notes
- Ensure that every angle and length adheres to the conditions provided.
- Use a ruler and protractor for accuracy to ensure the measurements and angles are correct.
You should now have a quadrilateral \(PQRS\) constructed according to the given specifications.