For a triangle ABC, find the measure of AB given m 55° 55°
55
°
55\degree , m 44° 44°
44
°
44\degree , and side b=68. Hint: this is not a right triangle so you will need to use either the law of sines or the law of cosines to solve it. These formulas are on your Geometry Reference Sheet.
To find the measure of side AB, we can use the Law of Cosines.
The Law of Cosines states:
c^2 = a^2 + b^2 - 2ab cos(C)
Where:
- a, b, and c are the sides of the triangle
- A, B, and C are the angles opposite to the sides a, b, and c, respectively
Since we want to find the measure of side AB (which is opposite to angle C), we have:
c^2 = a^2 + b^2 - 2ab cos(C)
AB^2 = AC^2 + BC^2 - 2(AC)(BC) cos(55°)
Given that AC = 68 and BC = 68, we can substitute these values into the formula:
AB^2 = 68^2 + 68^2 - 2(68)(68) cos(55°)
AB^2 = 4624 + 4624 - 2(4624) cos(55°)
AB^2 = 9248 - 9248 cos(55°)
Now, we can calculate AB using a calculator:
AB ≈ √(9248 - 9248 cos(55°))
AB ≈ √(9248 - 9248*0.5736)
AB ≈ √(9248 - 5301.5168)
AB ≈ √3946.4832
AB ≈ 62.79
Therefore, the measure of side AB is approximately 62.79.