Construct a triangle PQR,where angle QPR=135degrees, PQ=8.4cm and QR=12.5cm. state length PR.
Bisect
Draw a base line PQ,draw a perpendicular line to
To construct triangle PQR with the given information, follow these steps:
Step 1: Draw a straight line segment PQ of length 8.4 cm.
Step 2: Choose a point P on this line segment.
Step 3: Place the sharp end of a compass at point P and draw an arc with a radius of 12.5 cm.
Step 4: Without changing the compass width, place the sharp end of the compass at point Q on the line segment PQ and draw another arc.
Step 5: The intersection of the two arcs will give you point R. Connect points P, Q, and R to form triangle PQR.
To find the length of PR, you can use the Pythagorean theorem or the Law of Cosines since you know the lengths of two sides and the measure of one angle.
Using the Law of Cosines, the formula is:
PR^2 = PQ^2 + QR^2 - 2 * PQ * QR * cos(QPR)
We can substitute the given values:
PR^2 = 8.4^2 + 12.5^2 - 2 * 8.4 * 12.5 * cos(135°)
To find the value of cos(135°), we can use the fact that cos(135°) = -cos(45°). Since cos(45°) = sqrt(2)/2, we have:
cos(135°) = -cos(45°) = -sqrt(2)/2
Substituting the values, we get:
PR^2 = 8.4^2 + 12.5^2 - 2 * 8.4 * 12.5 * (-sqrt(2)/2)
Solving this equation will give us the value of PR.
To find the length of PR in triangle PQR, we can use the Law of Cosines. The Law of Cosines states that in any triangle, the square of one side is equal to the sum of the squares of the other two sides minus two times the product of the lengths of those two sides multiplied by the cosine of the included angle.
In this case, we are given PQ = 8.4 cm and QR = 12.5 cm, and we need to find PR.
The formula to find PR using the Law of Cosines is:
PR² = PQ² + QR² - 2 * PQ * QR * cos(QPR)
First, let's find the value of cosine of angle QPR. Since the angle QPR = 135 degrees, we can substitute this value into the equation:
PR² = 8.4² + 12.5² - 2 * 8.4 * 12.5 * cos(135)
Next, we need to find the cosine of 135 degrees. In most scientific calculators, you can find the cosine of an angle by pressing the "cos" button followed by the angle value. So, calculate the cosine of 135 degrees:
cos(135) ≈ -0.707
Now we can substitute this value into the equation:
PR² = 8.4² + 12.5² - 2 * 8.4 * 12.5 * (-0.707)
Simplifying this equation, we get:
PR² ≈ 70.56 + 156.25 + 170.94 ≈ 397.75
To find the value of PR, we need to take the square root of both sides:
PR ≈ √397.75 ≈ 19.94 cm.
Therefore, the length of PR is approximately 19.94 cm.