The longest side of a triangle are 13 ft., 15 ft., and 11 ft.. Find the measure of the opposite longest side. Round your answer to the nearest degree.
A. 46 degrees
B. 58 degrees
C. 77 degrees
D. 103 degrees
rip dude
Well, it seems we have a "longest side" contest in this triangle! Let's figure out who takes the crown.
Now, to find the measure of the opposite longest side, we need to use the Law of Cosines. According to this law, the square of the longest side (c) is equal to the sum of the squares of the other two sides (a and b), minus two times the product of those sides multiplied by the cosine of the angle (C) opposite the longest side. Phew!
We have the values of a, b, and c; they are 13 ft., 15 ft., and 11 ft., respectively. Now, let's find the angle C. Using the Law of Cosines, we can solve for C.
c² = a² + b² - 2ab * cos(C)
Substituting the given values:
11^2 = 13^2 + 15^2 - 2 * 13 * 15 * cos(C)
121 = 169 + 225 - 390 * cos(C)
121 = 394 - 390 * cos(C)
390 * cos(C) = 394 - 121
390 * cos(C) = 273
cos(C) = 273/390
cos(C) ≈ 0.7
Now, we just need to find the inverse cosine of 0.7 to determine the measure of angle C.
C ≈ 46 degrees
So, the measure of the opposite longest side is approximately 46 degrees. Therefore, the correct answer is A. 46 degrees. Congratulations to the winner of the "longest side" competition!
To solve this problem, we need to use the Law of Cosines, which states that for a triangle with sides a, b, and c, and angle C opposite side c:
c^2 = a^2 + b^2 - 2ab * cos(C)
In this case, the longest side of the triangle is 15 ft. Let's call this side c. The other two sides are 13 ft and 11 ft. Let's call these sides a and b, respectively.
Using the Law of Cosines, we can solve for angle C:
15^2 = 13^2 + 11^2 - 2 * 13 * 11 * cos(C)
225 = 169 + 121 - 286 * cos(C)
Now, rearranging the equation to solve for cos(C):
225 - 169 - 121 = -286 * cos(C)
-65 = -286 * cos(C)
Simplifying the equation:
cos(C) = -65 / -286
cos(C) ≈ 0.2273
To find the measure of angle C, we need to take the arc cosine (inverse cosine) of 0.2273:
C ≈ cos^(-1)(0.2273)
Using a calculator, we find that C ≈ 77.22 degrees. Rounded to the nearest degree, this is approximately 77 degrees.
Therefore, the measure of the opposite longest side is approximately 77 degrees.
So, the correct answer is C. 77 degrees.
To find the measure of the opposite longest side in a triangle, we can use the law of cosines. The law of cosines states that c^2 = a^2 + b^2 - 2ab*cos(C), where c is the length of the longest side, a and b are the lengths of the other two sides, and C is the angle opposite the longest side.
In this case, the longest side is 15 ft. Let's call the other two sides a = 11 ft. and b = 13 ft. Now we can plug these values into the formula:
c^2 = a^2 + b^2 - 2ab*cos(C)
15^2 = 11^2 + 13^2 - 2(11)(13)*cos(C)
225 = 121 + 169 - 286*cos(C)
225 = 290 - 286*cos(C)
To find cos(C), we need to rearrange the equation:
286*cos(C) = 290 - 225
286*cos(C) = 65
Now, we can solve for cos(C):
cos(C) = 65 / 286
To find the measure of angle C, we can take the inverse cosine (arccos) of cos(C) using a calculator. Round the answer to the nearest degree:
C ≈ arccos(65 / 286) ≈ 77 degrees
Therefore, the measure of the opposite longest side is approximately 77 degrees.
The correct answer is:
C. 77 degrees
I'll disregard the garbled language, and just note that the largest angle will be opposite the longest side. Thus, it will be between the two shortest sides. So, using the law of cosines,
15^2 = 11^2 + 13^2 - 2*11*13 cosθ