Find the measure of the largest acute angle in a 5-12-13 triangle.
You should recognize this to be a right-angled triangle, since
5^2 + 12^2 = 13^2
tanØ = 12/5 = 2.4 , where Ø is the angle opposite the 12 side
Ø = appr 67.4°
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Well, let me put on my thinking wig, or should I say, my funny wig. In a 5-12-13 triangle, we have sides of length 5, 12, and 13. Now, I don't want to be obtuse, but there's actually only one acute angle in a triangle. If I were to measure it, it would be approximately 22.62 degrees. But remember, this is just an estimate, so don't take it too trigonometrically!
To find the measure of the largest acute angle in a triangle with sides 5, 12, and 13, we can use the Law of Cosines.
The Law of Cosines states:
c^2 = a^2 + b^2 - 2ab * cos(C),
Where c represents the side opposite the angle C, and a and b represent the other two sides.
In this case, we have:
c = 13,
a = 5,
b = 12,
Let's plug these values into the formula:
13^2 = 5^2 + 12^2 - 2(5)(12) * cos(C).
169 = 25 + 144 - 120 * cos(C).
169 = 169 - 120 * cos(C).
0 = -120 * cos(C).
We can divide both sides of the equation by -120:
0 = cos(C).
The only way for the cosine of an angle to be 0 is if the angle is 90 degrees (or pi/2 radians).
Therefore, the largest acute angle in the 5-12-13 triangle is 90 degrees.
To find the measure of the largest acute angle in a triangle, we first need to determine which angle is the largest. In a right triangle, the largest angle is always the right angle which measures 90 degrees. However, in this case, we are dealing with an acute triangle which means all angles are less than 90 degrees.
In a 5-12-13 triangle, the sides are proportional to each other, with the ratio being 1:2:5. Therefore, the largest side is the side opposite the largest angle. In this case, the side length of 13 corresponds to the largest side.
To find the corresponding angle, we can use the Law of Cosines, which states that for any triangle with sides a, b, and c and opposite angles A, B, and C respectively, the following equation holds true:
c^2 = a^2 + b^2 - 2ab * cos(C)
In our case, we have a 5-12-13 triangle and we want to find angle C, which is opposite the side of length 13. Plugging in the values, we get:
13^2 = 5^2 + 12^2 - 2(5)(12) * cos(C)
169 = 25 + 144 - 120 * cos(C)
169 = 169 - 120 * cos(C)
120 * cos(C) = 0
cos(C) = 0
Now, we need to find the value of C such that cos(C) equals 0. Since cosine is equal to 0 at 90-degree intervals, we can conclude that C is a right angle or 90 degrees.
In conclusion, the measure of the largest acute angle in a 5-12-13 triangle is 90 degrees.