Answer:
11x=0
Step-by-step explanation:
Theorem: When two secants meet at an external point, the products of the secant and the external segment are equal.
In essence, we are saying that QR × PR = TR × SR
Thus, 16 × 6 = 8(x + 8)
96 = 8(x + 8)
Dividing both sides by 8, we produce:
12 = x + 8
x = 4
Thus, we can conclude that the missing side, x, is 4.
![\bar{x} = 0](https://tex.z-dn.net/?f=%5Cbar%7Bx%7D%20%3D%200)
![\bar{y} =\dfrac{136}{125}](https://tex.z-dn.net/?f=%5Cbar%7By%7D%20%3D%5Cdfrac%7B136%7D%7B125%7D)
Step-by-step explanation:
Let's define our functions
as follows:
![f(x) = x^2 + 1](https://tex.z-dn.net/?f=f%28x%29%20%3D%20x%5E2%20%2B%201)
![g(x) = 6x^2](https://tex.z-dn.net/?f=g%28x%29%20%3D%206x%5E2)
The two functions intersect when
and that occurs at
so they're going to be the limits of integration. To solve for the coordinates of the centroid
, we need to solve for the area A first:
![\displaystyle A = \int_a^b [f(x) - g(x)]dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%20A%20%3D%20%5Cint_a%5Eb%20%5Bf%28x%29%20-%20g%28x%29%5Ddx)
![\displaystyle \:\:\:\:\:\:\:=\int_{-\frac{1}{5}}^{+\frac{1}{5}}[(x^2 + 1) - 6x^2]dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%3D%5Cint_%7B-%5Cfrac%7B1%7D%7B5%7D%7D%5E%7B%2B%5Cfrac%7B1%7D%7B5%7D%7D%5B%28x%5E2%20%2B%201%29%20-%206x%5E2%5Ddx)
![\displaystyle \:\:\:\:\:\:\:=\int_{-\frac{1}{5}}^{+\frac{1}{5}}(1 - 5x^2)dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%3D%5Cint_%7B-%5Cfrac%7B1%7D%7B5%7D%7D%5E%7B%2B%5Cfrac%7B1%7D%7B5%7D%7D%281%20-%205x%5E2%29dx)
![\displaystyle \:\:\:\:\:\:\:=\left(x - \frac{5}{3}x^3 \right)_{-\frac{1}{5}}^{+\frac{1}{5}}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%3D%5Cleft%28x%20-%20%5Cfrac%7B5%7D%7B3%7Dx%5E3%20%5Cright%29_%7B-%5Cfrac%7B1%7D%7B5%7D%7D%5E%7B%2B%5Cfrac%7B1%7D%7B5%7D%7D)
![\:\:\:\:\:\:\:= \dfrac{28}{75}](https://tex.z-dn.net/?f=%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%3D%20%5Cdfrac%7B28%7D%7B75%7D)
The x-coordinate of the centroid
is given by
![\displaystyle \bar{x} = \dfrac{1}{A}\int_a^b x[f(x) - g(x)]dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cbar%7Bx%7D%20%3D%20%5Cdfrac%7B1%7D%7BA%7D%5Cint_a%5Eb%20x%5Bf%28x%29%20-%20g%28x%29%5Ddx)
![\displaystyle \:\:\:\:\:\:\:= \frac{75}{28}\int_{-\frac{1}{5}}^{+\frac{1}{5}} (x - 5x^3)dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%3D%20%5Cfrac%7B75%7D%7B28%7D%5Cint_%7B-%5Cfrac%7B1%7D%7B5%7D%7D%5E%7B%2B%5Cfrac%7B1%7D%7B5%7D%7D%20%28x%20-%205x%5E3%29dx)
![\:\:\:\:\:\:\:=\dfrac{75}{28}\left(\dfrac{1}{2}x^2 -\dfrac{5}{4}x^4 \right)_{-\frac{1}{5}}^{+\frac{1}{5}}](https://tex.z-dn.net/?f=%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%3D%5Cdfrac%7B75%7D%7B28%7D%5Cleft%28%5Cdfrac%7B1%7D%7B2%7Dx%5E2%20-%5Cdfrac%7B5%7D%7B4%7Dx%5E4%20%5Cright%29_%7B-%5Cfrac%7B1%7D%7B5%7D%7D%5E%7B%2B%5Cfrac%7B1%7D%7B5%7D%7D)
![\:\:\:\:\:\:\:= 0](https://tex.z-dn.net/?f=%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%3D%200)
The y-coordinate of the centroid
is given by
![\displaystyle \bar{y} = \frac{1}{A}\int_a^b \frac{1}{2}[f^2(x) - g^2(x)]dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cbar%7By%7D%20%3D%20%5Cfrac%7B1%7D%7BA%7D%5Cint_a%5Eb%20%5Cfrac%7B1%7D%7B2%7D%5Bf%5E2%28x%29%20-%20g%5E2%28x%29%5Ddx)
![\displaystyle \:\:\:\:\:\:\:=\frac{75}{28}\int_{-\frac{1}{5}}^{+\frac{1}{5}} \frac{1}{2}(-35x^4 + 2x^2 + 1)dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%3D%5Cfrac%7B75%7D%7B28%7D%5Cint_%7B-%5Cfrac%7B1%7D%7B5%7D%7D%5E%7B%2B%5Cfrac%7B1%7D%7B5%7D%7D%20%5Cfrac%7B1%7D%7B2%7D%28-35x%5E4%20%2B%202x%5E2%20%2B%201%29dx)
![\:\:\:\:\:\:\:=\frac{75}{56} \left[-7x^5 + \frac{2}{3}x^3 + x \right]_{-\frac{1}{5}}^{+\frac{1}{5}}](https://tex.z-dn.net/?f=%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%3D%5Cfrac%7B75%7D%7B56%7D%20%5Cleft%5B-7x%5E5%20%2B%20%5Cfrac%7B2%7D%7B3%7Dx%5E3%20%2B%20x%20%5Cright%5D_%7B-%5Cfrac%7B1%7D%7B5%7D%7D%5E%7B%2B%5Cfrac%7B1%7D%7B5%7D%7D)
![\:\:\:\:\:\:\:=\dfrac{136}{125}](https://tex.z-dn.net/?f=%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%5C%3A%3D%5Cdfrac%7B136%7D%7B125%7D)
Step-by-step explanation:
AB = AC ( Given)
angle BAD = angle DAC. (AD bisects BAC)
AD = AD. ( common side )
abd is congruent to adc. ( SAS Axiom)