Angle QRS is a straight angle, so a+b+c+d = 180. If a=2c, c=2b, and b=d/3, what are the values of a,b,c,and d?
You have 4 equations in 4 unknowns. Use substitution to reduce the number of variables.
2c + c/2 + c + 3b = 7c/2 + 3c/2 = 5c = 180
c = 180/5 = 36 degrees
a = 2c = 72
b = c/2 = 18
d = 3b = 54
They add up to 180, as required.
To solve for the values of a, b, c, and d, we need to use the given information and the equation a + b + c + d = 180.
Given:
a = 2c
c = 2b
b = d/3
We can substitute these values into the equation to solve for the variables:
a + b + c + d = 180
Using the given information:
2c + b + c + d = 180
Substituting c = 2b and b = d/3:
2(2b) + (d/3) + 2b + d = 180
Simplifying:
4b + (d/3) + 2b + d = 180
Combining like terms:
6b + (4d/3) = 180
To eliminate fractions, we can multiply through by the least common denominator, which is 3:
3(6b) + 3(4d/3) = 3(180)
Simplifying:
18b + 4d = 540
Now we have a system of two linear equations:
2c + b + c + d = 180
18b + 4d = 540
To solve this system of equations, we can use substitution or elimination methods.
Let's use substitution:
From c = 2b, we can substitute c with 2b in the first equation:
2(2b) + b + 2b + d = 180
4b + b + 2b + d = 180
7b + d = 180 (eq.1)
Now, let's isolate d in terms of b from the second equation:
18b + 4d = 540
4d = 540 - 18b
d = (540 - 18b)/4
d = (135 - 4.5b) (eq.2)
Now, substitute eq.2 into eq.1:
7b + (135 - 4.5b) = 180
7b - 4.5b = 180 - 135
2.5b = 45
b = 45/2.5
b = 18
Substitute the value of b back into eq.1:
7(18) + d = 180
126 + d = 180
d = 180 - 126
d = 54
Substitute the values of b and d back into eq.2:
a = 2c = 2(2b) = 2(2 * 18) = 2(36) = 72
c = 2b = 2 * 18 = 36
Therefore, the values of a, b, c, and d are:
a = 72
b = 18
c = 36
d = 54