The line QS bisects <PQR, m<PQR = (x2), and m<PQS = (2x +6). Find all the possible measures for <PQR.
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PQR = X^2.
PQS = 2x + 6.
PQS = PQR/2,
2x + 6 = X^2/2,
Multiply both sides by 2:
4x + 12 = X^2,
x^2 - 4x - 12 = 0,
(x+2)(x-6) = 0,
x+2 = 0,
X = -2.
x-6 = 0,
X = 6.
PQR = X^2 = (-2)^2 = 4.
PQR = 6^2 = 36.
To find all the possible measures for angle PQR, we need to use the properties of angles formed by a bisecting line.
We're given that line QS bisects angle PQR. This means that the measure of angle PQS is equal to the measure of angle SQR. So, we can conclude that:
m<PQS = m<SQR
Now, we're given that m<PQR = (x^2) and m<PQS = (2x + 6).
Since the measure of angle PQS is equal to the measure of angle SQR, we can equate these two measures:
x^2 = 2x + 6
To find all the possible measures of angle PQR, we need to solve this quadratic equation for x.
Rearranging the equation, we have:
x^2 - 2x - 6 = 0
Now, we can use factoring, completing the square, or the quadratic formula to solve for x. Let's use the quadratic formula.
The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the solutions for x can be found using the formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation x^2 - 2x - 6 = 0, we have a = 1, b = -2, and c = -6.
Plugging these values into the quadratic formula, we have:
x = (-(-2) ± √((-2)^2 - 4(1)(-6))) / (2(1))
Simplifying further, we get:
x = (2 ± √(4 + 24)) / 2
x = (2 ± √28) / 2
x = (2 ± 2√7) / 2
We can simplify this expression by canceling the common factors in the numerator and denominator:
x = 1 ± √7
So, we have two possible values for x:
x = 1 + √7
x = 1 - √7
Now, we can substitute these values back into the equation m<PQR = x^2 to find the possible measures for angle PQR.
If x = 1 + √7, then:
m<PQR = (1 + √7)^2 = 1 + 2√7 + 7 = 8 + 2√7
If x = 1 - √7, then:
m<PQR = (1 - √7)^2 = 1 - 2√7 + 7 = 8 - 2√7
Therefore, the possible measures for angle PQR are:
m<PQR = 8 + 2√7 and m<PQR = 8 - 2√7.