Triangle ABC has side lengths a,b and c. If these lengths satisfy a^2=a+2b+2c and −3=a+2b−2c, what is the measure (in degrees) of the largest angle?
To find the measure of the largest angle in Triangle ABC, we can use the Law of Cosines. The Law of Cosines relates the side lengths of a triangle with the cosine of one of its angles.
The Law of Cosines states that:
c^2 = a^2 + b^2 - 2ab * cos(C)
In this case, we have the following information:
a^2 = a + 2b + 2c (Equation 1)
-3 = a + 2b - 2c (Equation 2)
Let's solve these equations simultaneously to find the values of a, b, and c.
From Equation 1:
a^2 = a + 2b + 2c
Rearranging the equation:
a^2 - a = 2b + 2c
a(a - 1) = 2(b + c)
From Equation 2:
-3 = a + 2b - 2c
Since we have two equations with two variables (a and b), we can substitute the value of a from Equation 2 into Equation 1:
(a - 1) (a) = 2(b + c)
-3(a - 1) = 2b + 2c
Expanding the equation:
-3a + 3 = 2b + 2c
We now have two equations with two variables:
-3 = 2b + 2c (Equation 3)
-3a + 3 = 2b + 2c (Equation 4)
Let's solve these equations simultaneously:
From Equation 3:
-3 = 2b + 2c
Divide through by 2:
-1.5 = b + c (Equation 5)
Substitute b + c from Equation 5 into Equation 4:
-3a + 3 = 2b + 2c
-3a + 3 = 2(-1.5)
-3a + 3 = -3
Rearranging the equation, we find the value of a:
-3a = -3 - 3
-3a = -6
a = 2
Substitute the value of a back into Equation 3:
-3 = 2b + 2c
-3 = 2b + 2(-1.5)
-3 = 2b - 3
2b = 0
b = 0
Now that we have the values of a and b, we can substitute them and solve for c:
From Equation 1:
a^2 = a + 2b + 2c
Plugging in the values of a and b, we get:
4 = 2 + 2(0) + 2c
4 = 2 + 2c
2c = 2
c = 1
Now that we have the values of a, b, and c, we can use the Law of Cosines to find the measure of the angle opposite the largest side (c). From the Law of Cosines:
c^2 = a^2 + b^2 - 2ab * cos(C)
Plugging in the values we found:
1^2 = 2^2 + 0^2 - 2(2)(0) * cos(C)
1 = 4 - 0 * cos(C)
cos(C) = (4 - 1) / (2 * 2)
cos(C) = 3 / 4
To find the measure of angle C, we can take the inverse cosine (cos^-1) of 3/4:
C = cos^-1(3/4)
Using a calculator, we find:
C ≈ 41.4 degrees
Therefore, the measure of the largest angle in Triangle ABC is approximately 41.4 degrees.