To find the bisector of angle CBA in triangle CBA, we need to first determine the slopes of lines CA and CB, and then find the perpendicular bisector of the side AB.
1. **Slope of CA (Line CD):**
Given that point C lies on the y-axis (C(0, c)) and point D is (6, 1), we can find the slope of CA (CD) using the formula:
Slope (m_CD) = (y2 - y1) / (x2 - x1) = (1 - c) / (6 - 0) = (1 - c) / 6
2. **Slope of CB (Line CE):**
Given that point E is (9, 3) and point C is on the y-axis (C(0, c)), we can find the slope of CB (CE) using the equation given for CB:
7y = x + 14
We can rewrite this in slope-intercept form (y = mx + b):
y = (1/7)x + 2
So, the slope of CB (CE) is 1/7.
Now, to find the bisector of angle CBA, we need to find a line that is perpendicular to both CA (CD) and CB (CE) and passes through point B(0, -24), which is the intersection of lines AB and CB.
**Slope of Bisector (m_bisector):**
The product of the slopes of two perpendicular lines is -1. Therefore:
m_CD * m_bisector = -1
(1 - c) / 6 * m_bisector = -1
Now, solve for m_bisector:
m_bisector = -6 / (1 - c)
**Slope of CB and Bisector (m_CE and m_bisector):**
Since line CB (CE) has a slope of 1/7 and the bisector must be perpendicular to it, we have:
m_bisector * m_CE = -1
(-6 / (1 - c)) * (1/7) = -1
Now, solve for (1 - c):
(1 - c) = (-6 / (1 - c)) * (1/7)
Now, cross-multiply:
7(1 - c) = -6
Distribute on the left side:
7 - 7c = -6
Now, isolate -7c on the left side:
-7c = -6 - 7
-7c = -13
Finally, solve for c:
c = -13 / -7
c = 13 / 7
So, the y-coordinate of point C is 13/7.
Now, we have the slope of the bisector (m_bisector) as -6 / (1 - c), where c = 13/7:
m_bisector = -6 / (1 - 13/7)
m_bisector = -6 / (-6/7)
m_bisector = -7
Therefore, the slope of the bisector of angle CBA is -7. Now you can find the equation of the bisector that passes through point B(0, -24) using the point-slope form:
y - y1 = m(x - x1)
y - (-24) = -7(x - 0)
y + 24 = -7x
This is the equation of the bisector of angle CBA.