Using a ruler and a pair of compasses only,construct a quadrilateral PQRS,given that /PQ/=8cm,PQS=30°,/PR/=9cm,/QS/=10cm and/RS/=/RQ/.Measure(a)QRS(b)/RS/
You will have to know how to draw a 60° angle, by constructing an equilateral triangle.
Once you have that, bisect the angle to get your 30° angle.
After that it is just a matter of measuring the sides with your ruler
To construct the quadrilateral PQRS using a ruler and a pair of compasses, follow these step-by-step instructions:
Step 1: Draw a line segment PQ with a length of 8 cm using a ruler.
Step 2: With point P as the center, draw an arc using the compasses to create an arc above the line segment PQ. This arc should be larger than the length of PR.
Step 3: Without changing the compasses' width, place the compasses' point on point Q and draw another arc to intersect the previously drawn arc. Label this point of intersection as R.
Step 4: Join points P and R with a straight line segment using a ruler.
Step 5: With point R as the center, draw an arc using the compasses with a width of 9 cm to cut the line segment PR, creating a point of intersection above PR. Label this point as S.
Step 6: Join points Q and S with a straight line segment using a ruler.
You have now constructed the quadrilateral PQRS.
To measure the angles and sides, follow these additional steps:
(a) To measure ∠QRS (angle QRS):
- Use a protractor to measure the angle formed by lines QS and RS.
(b) To measure |RS| (length of RS):
- Use a ruler to measure the distance between points R and S.
Keep in mind that measuring these values accurately will depend on the precision of your tools and drawing.
To construct the quadrilateral PQRS using a ruler and a pair of compasses only, follow these steps:
Step 1: Draw a line segment PQ of length 8 cm using the ruler.
Step 2: With the center at point P, open your compass to a convenient radius and draw an arc that intersects line segment PQ. Label the point of intersection as A.
Step 3: With the center at point Q, open your compass to a radius equal to the length of line segment QS (10 cm) and draw an arc that intersects line segment PQ. Label the point of intersection as S.
Step 4: With the center at point S, open your compass to a radius equal to the length of line segment RS (assumed to be 9 cm as /RS/=/RQ/) and draw an arc that intersects line segment PQ. Label the point of intersection as R.
Step 5: Connect points R and P to complete the quadrilateral PQRS.
To measure the angles and sides of the quadrilateral:
(a) To measure angle QRS:
- Use the ruler to measure the length of line segment QR. The length of QR should be equal to RS (9 cm), as given in the question.
- Use the compass to measure the angle formed between line segments QR and RS (angle QRS). Place the compass with one leg on point R and the other leg on point Q, then read the angle measurement on the compass.
(b) To measure the length of side RS:
- Use the ruler to measure the length of line segment RS. The length of RS should be the same as the length of QR, which is 9 cm as given in the question.