construct a triangle PQR in which angle Q = 60 degree , angle R = 75 degree and the sum of three sides is 12 cm
We will need to find the sides
angle P = 180 - 60 - 75 = 45°
let PQ=x , QR = y , PR = z
x/sin75 = y/sin45 = z/sin60
let's assign an arbitrary value of y = 1 to our smallest side
using x/sin75 = 1/sin45
we get x = .1.366025
using 1/sin45 = z/sin60
z = 1.224745
so sum of those sides
= 1 + 1.366025+1.224745
= 3.59077
but we want this sum to be 12, so each of our sides must be multiplied by a factor of 12/3.59077 = 3.3419
giving me sides of
x = 4.565122
y = 3.3419
z = 4.09298
Now that you have the sides, round them off to suitable values, and construct your triangle
To construct a triangle PQR with angle Q = 60 degrees, angle R = 75 degrees, and the sum of three sides equal to 12 cm, we will follow these steps:
Step 1: Draw a line segment PQ of any convenient length. This will be one of the sides of our triangle.
Step 2: At point P, draw a ray in any direction and label the endpoint as R. This will be another side of our triangle.
Step 3: Measure an angle of 60 degrees at point Q using a protractor. This will be the angle Q in our triangle.
Step 4: Using a protractor, measure an angle of 75 degrees at point R on the ray. This will be the angle R in our triangle.
Step 5: To complete the triangle, we need to determine the length of the third side. Since the sum of three sides is given as 12 cm, we can use this information to find the length of the third side.
To do this, 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 those sides multiplied by the cosine of the included angle.
In our case, the unknown side length can be labeled as x. So we have:
x^2 = PQ^2 + PR^2 - 2(PQ)(PR)cos(75)
We know PQ and PR from the construction, so we can substitute their values into the equation.
Step 6: Solve the equation to find the value of x, which will give us the length of the third side QR.
Once you have determined the length of QR, you can draw a line segment from Q to R with the measured length to complete the triangle PQR.
Please note that the accuracy of the construction and measurements will affect the exact dimensions and appearance of the triangle.