If KL=3cm,KM=7cm and LM=5cm.Calculate the angle K,M and the area of angle KLM
Use Law of Cosines.
CosK = (l^2 + m^2 -k^2) / 2lm,
CosK = (&^2 + 3^2 - 5^2) / 2*7*3,
1. CosK = 20 / 42 = 0.47619,
K = 61.56 Deg.
2. sinK/k = sinM/m,
sin61.56/5 = sinM/3,
5*sinM = 3*sin61.56,
Divide both sides by 5:
sinM = 0.6*sin61.56 = 0.52759,
M = 31.84 Deg.
3. L = 180 -61.56 - 31.84 = 86.6 Deg.
4. Area = 0.5*l*m*sinK,
Area = 0.5*7*3*sin61.56 = 9.23cm^2.
Correction:CosK=(7^2 + 3^2 - 5^2)/2*7*3
To calculate the angle KLM, we can use the law of cosines, which states that in a triangle, the square of one side is equal to the sum of the squares of the other two sides minus two times their product, multiplied by the cosine of the included angle.
Let's label the angle KLM as angle A.
Using the law of cosines, we have:
LM^2 = KL^2 + KM^2 - 2(KL * KM * cos(A))
Substituting the given values, we have:
5^2 = 3^2 + 7^2 - 2(3 * 7 * cos(A))
Simplifying further:
25 = 9 + 49 - 42cos(A)
25 = 58 - 42cos(A)
42cos(A) = 58 - 25
42cos(A) = 33
cos(A) = 33/42
To find angle A, we can take the inverse cosine (cos^-1) of both sides:
A = cos^-1(33/42)
Now, we can use a calculator or reference table to find the value of cos^-1(33/42) which is approximately 43.55 degrees.
So, angle KLM ≈ 43.55 degrees.
To calculate the area of angle KLM, we use the formula for the area of a triangle, which is 1/2 * base * height.
The base of angle KLM is KM, which is 7 cm.
The height of angle KLM can be found by dropping a perpendicular from point M to line KL. Let's call the point of intersection X.
To find the height, we can use the Pythagorean theorem.
Since we have a right triangle KMX, we can say:
KM^2 = KX^2 + MX^2
Substituting the values, we have:
7^2 = KX^2 + MX^2
Simplifying further, we have:
49 = KX^2 + MX^2
Since point X is on line KL, and KM is perpendicular to KL, MX = LM = 5 cm.
Substituting the value:
49 = KX^2 + 5^2
49 = KX^2 + 25
KX^2 = 49 - 25
KX^2 = 24
KX = sqrt(24)
KX ≈ 4.899 cm
Now, we have the height of the triangle, KX, which is approximately 4.899 cm.
Using the formula for the area of a triangle:
Area = 1/2 * base * height
Area = 1/2 * 7 * 4.899
Area ≈ 17.1475 cm²
So, the area of angle KLM ≈ 17.1475 cm².