construct a quadrilateral PQRS such that /PQ/=6cm,/QS/=/RS/=9,PQS=75° find /PS/and /QR/.
use the law of cosines to find PS:
PS^2 = 6^2 + 9^2 - 2*6*9*cos75°
PS ≈ 9.44
As for QR, that's a bit harder. There is no way to say just where point R is. All you know is that it lies on a circle with center at S, with radius 9.
Try drawing PQRS and you will see that you need a bit more information.
To construct quadrilateral PQRS with the given information, follow these step-by-step instructions:
Step 1: Draw a line segment of length 6 cm and label it PQ.
Step 2: At point P, draw an angle of 75 degrees. This will be angle PQS.
Step 3: Using a protractor, draw a line segment of length 9 cm at angle PQS and label it QS.
Step 4: At point Q, draw another angle equal to 75 degrees. This will be angle QSR.
Step 5: Using a protractor, draw a line segment of length 9 cm at angle QSR and label it RS.
Step 6: Join point P and R to form line segment PR.
Step 7: Point S will be the intersection of line QS and line RS.
Step 8: Label line segment PS and QR.
Now, to find the lengths of PS and QR:
Since we have a quadrilateral, we can use the property that opposite sides of a quadrilateral are equal in length.
Therefore, PS = QR = 9 cm.
To construct quadrilateral PQRS, follow these steps:
1. Draw a straight line segment PQ of length 6 cm.
2. At point P, make an angle of 75 degrees. This will be angle PQS.
3. From point Q, draw a ray in the direction opposite to SP (away from P). Make sure the length of QS is 9 cm.
4. At point S, draw a ray in the direction opposite to RQ (away from Q). Make sure the length of RS is also 9 cm.
5. Finally, draw RS to complete the quadrilateral PQRS.
To find the lengths of PS and QR, we can use the Law of Cosines. This law states that in any triangle, the square of one side is equal to the sum of the squares of the other two sides, minus twice the product of the two sides and the cosine of the included angle.
In our case, we have triangle PQS with sides PQ = 6 cm, QS = 9 cm, and angle PQS = 75 degrees.
Using the Law of Cosines, we can calculate the length of PS:
PS^2 = PQ^2 + QS^2 - 2 * PQ * QS * cos(PQS)
Substituting the values into the equation:
PS^2 = 6^2 + 9^2 - 2 * 6 * 9 * cos(75°)
Once you calculate the value of PS^2, take the square root to find the length of PS.
Similarly, to find the length of QR, you can use the Law of Cosines on triangle QRS, with sides QS = 9 cm, RS = 9 cm, and angle QRS = 180° - PQS = 180° - 75° = 105°.