Using ruler and pair of compasses only, construct a quadrilateral PQRS,such that/PQ/ =8cm ,QPS=105 , PQS=30, /PR/=9cm, and /RS/ =/RQ/
Draw PQ of length 8
Construct the two angles. Note that 105 = 60+45, 30 = 60/2. If you can construct a 45 or 60-degree angle, this will be easy. Where the line from Q intersects the line from P will be S.
Draw QS
Since PR=RS, R lies on the perpendicular bisector of QS. Construct that line segment.
Draw a circle of radius 9 with center at P. Where it intersects the perpendicular bisector of QS will be R.
google can help if you have difficulty with any of these steps.
To construct the quadrilateral PQRS using only a ruler and pair of compasses, follow these steps:
1. Draw a line segment PQ of length 8 cm using a ruler.
2. Use the compass to mark a point P on the line segment PQ.
3. With the same compass width, place the compass at point P and draw an arc above the line segment PQ.
4. Without changing the compass width, place the compass at point Q and draw another arc intersecting the first arc. Label this point of intersection as S.
5. Use the ruler to draw a line segment RS connecting points R and S.
6. With the same compass width, place the compass at point R and draw an arc below the line segment RS.
7. Without changing the compass width, place the compass at point S and draw another arc intersecting the first arc. Label this point of intersection as Q.
8. Use the ruler to draw a line segment PQ connecting points P and Q. You have now completed the construction of the quadrilateral PQRS.
To verify if the construct satisfies the given conditions:
- Measure the length of PQ with a ruler; it should measure 8 cm.
- Measure the angle QPS using a protractor; it should measure 105 degrees.
- Measure the angle PQS using a protractor; it should measure 30 degrees.
- Measure the length of PR with a ruler; it should measure 9 cm.
- Measure the length of RS and RQ with a ruler; both should be equal.
That's it! You have successfully constructed the quadrilateral PQRS using only a ruler and pair of compasses.