Question

# In ΔKLM, k = 56 inches, m = 51 inches and ∠M=28°. Find all possible values of ∠K, to the nearest degree.

Answer

The possible values of ∠K, to the nearest degree, are 48° and 89°.

- Q:What information do we have about the triangle? A:We know that KL = KM since they are two sides of the same triangle, and we know that ∠M = 28°, k=56 in and m=51 in.
- Q:What formula can we use to solve for ∠K? A:We can use the Law of Cosines, which states that c^2 = a^2 + b^2 - 2abcos(C), where c is the side opposite to angle C.
- Q:What is the value of c in this triangle? A:The side opposite to angle M is KL, which is equal to KM. Therefore, c = KL = KM = 51 in.
- Q:What are a and b in the triangle? A:a is the side opposite to angle K, which is k = 56 in, and b is the side opposite to angle L, which we can solve for using the Law of Cosines. A^2 = B^2 + C^2 - 2BC cos(A) --> L^2 = K^2 + M^2 - 2KMcos(L) --> L = sqrt(K^2 + M^2 - 2KMcos(L)), so b = L.
- Q:What is the value of cos(L)? A:cos(L) = (K^2 + M^2 - L^2)/(2KM) --> cos(L) = (56^2 + 51^2 - L^2)/(2*56*51) --> cos(L) = (6307 - L^2)/5712 --> L^2 = 6307 - 5712cos(L)
- Q:What are the possible values of L? A:Since cos(L) cannot be greater than 1 or less than -1, we have 6307 - 5712 <= L^2 <= 6307 + 5712, which gives 95 <= L^2 <= 12019. Therefore, 9.747 <= L <= 109.573, and we can round these values to the nearest degree to get the possible values of ∠L.