construct a quadrilateral PQRS such that PQ=8.5cm, QR=6cm PS=5cm. SPQ = 60 degrees. and PQR=105 degrees. construct a circle passing through, S, P, Q. measure the radius of the circle
To construct a quadrilateral PQRS and find the radius of the circle passing through points S, P, and Q, follow these steps:
Step 1: Draw a line segment PQ of 8.5 cm. This will be one side of the quadrilateral.
Step 2: From point P, construct an angle of 105 degrees. This can be done by using a protractor and measuring the desired angle from the reference line segment PQ.
Step 3: From the endpoint of line segment PQ, mark a point R on the constructed angle. The length of QR should be 6 cm, so measure and mark this length on the constructed angle.
Step 4: Draw a line segment RS connecting points R and S. Make sure this line is straight and not angled.
Step 5: From point S, construct an angle of 60 degrees. Again, using a protractor, measure the desired angle from the reference line segment RS.
Step 6: From the endpoint of line segment RS, mark a point Q on the constructed angle. The length of QS should be 5 cm, so measure and mark this length on the constructed angle.
Step 7: Connect points Q and P using a straight line segment to form the quadrilateral PQRS.
Step 8: To construct a circle passing through points S, P, and Q, find the circumcenter of the triangle SPQ. This can be done by drawing perpendicular bisectors of two sides of the triangle. In this case, the perpendicular bisectors of SP and PQ will intersect at the circumcenter.
Step 9: Using a compass, place the needle at one end of the perpendicular bisector of SP. Draw an arc that intersects the perpendicular bisector of PQ.
Step 10: Keep the compass radius unchanged and place the needle at the other end of the perpendicular bisector of SP. Draw another arc that intersects the same perpendicular bisector of PQ.
Step 11: The intersection point of these two arcs on the perpendicular bisector of PQ is the circumcenter of the triangle SPQ. Let's call this point O.
Step 12: From point O, use the compass to draw a circular arc passing through points S, P, and Q.
Step 13: Finally, measure the radius of the constructed circle using a ruler or a compass. Place the ruler from the center O to any point on the circle to measure the radius. This measurement will give you the radius of the circle passing through points S, P, and Q.