a skater skates along a straigh path from point A to point . Two streets that intersect a point C also intersect the path at points A and B. Street AC is 0.4 miles and street BC is 1.3 miles. The included angles at c is 107 . How far did the skater skate from point A to B
To find the distance the skater skate from point A to B, we need to use the Law of Cosines. This law relates the lengths of the sides of a triangle to the cosine of one of its angles.
In this case, we have a triangle with sides AC, BC, and AB, and an included angle at C.
The Law of Cosines states that:
c^2 = a^2 + b^2 - 2ab * cos(C)
Where c is the side opposite to angle C.
In our scenario, we want to find the distance AB, which is side c. We know the lengths of sides a (AC) and b (BC), as well as the measure of angle C.
Plugging in the values into the formula, we have:
AB^2 = AC^2 + BC^2 - 2 * AC * BC * cos(C)
Substituting the known values:
AB^2 = 0.4^2 + 1.3^2 - 2 * 0.4 * 1.3 * cos(107°)
Calculating the values inside the parentheses:
AB^2 = 0.16 + 1.69 - 0.208 * cos(107°)
Next, we need to find the cosine of 107°. You can use a calculator or a trigonometric table to determine the value of cos(107°).
Once you find the cosine value, multiply it by 0.208 and subtract it from 1.69.
Finally, add the values of 0.16 and the result from the previous step. Take the square root of this sum to find the distance AB.