A surveyor stakes out points A and B on sides of a building. Point C on the side of the building is 300 feet from A and 440 feet from B. Angle ACB measures 38 degrees. What is the distance from A to B?
Use the Law of Cosines to solve triangle ABC:
cosC = (a^2 + b^2 - c^2) / 2ab.
cos38=((440)^2+(300)^2-c^2)/2*440*300,
0.7880 = (283600 - c^2) / 264000,
Cross multiply:
-c^2 + 283600 = 208032,
-c^2 = 208032 - 283600,
-c^2 = - 75568,
c^2 = 75568,
c = 275ft. = Distance from A to B.
To find the distance from point A to B, we can use the Law of Cosines. Let's break down the steps:
Step 1: Label the sides of the triangle
- Let AB be the distance from A to B. This is the side we want to find.
- Let AC be 300 feet.
- Let BC be 440 feet.
Step 2: Apply the Law of Cosines
The Law of Cosines states that in a triangle with sides a, b, and c, and opposite angles A, B, and C respectively, the following equation holds:
c^2 = a^2 + b^2 - 2ab * cos(C)
In our case, we want to find the side AB, so we have:
AB^2 = AC^2 + BC^2 - 2 * AC * BC * cos(38°)
Step 3: Substitute the given values and solve for AB
- Substitute AC = 300 feet, BC = 440 feet, and the angle ACB = 38° into the equation:
AB^2 = 300^2 + 440^2 - 2 * 300 * 440 * cos(38°)
Simplifying the equation using basic arithmetic operations gives us:
AB^2 = 90000 + 193600 - 52800 * cos(38°)
AB^2 = 283600 - 52800 * cos(38°)
Step 4: Calculate AB
- Calculate the cosine of 38° (cos(38°)) using a calculator or any trigonometric tool.
- Multiply 52800 by the cosine value obtained in the previous step.
- Subtract the result from 283600.
- Finally, take the square root of the result to find the distance AB.
By following these steps, you can find the distance from point A to B in the given scenario.