Let's tackle the problem step by step to find the solution.
### (a) Find the equation of the line [tex]\( AC \)[/tex] in the form [tex]\( y = mx + c \)[/tex]
Given points [tex]\( A(-2, -3) \)[/tex] and [tex]\( C(1, 9) \)[/tex]:
1. Calculate the slope [tex]\( m \)[/tex]:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Substituting the given points [tex]\( (x_1, y_1) = (-2, -3) \)[/tex] and [tex]\( (x_2, y_2) = (1, 9) \)[/tex]:
[tex]\[
m = \frac{9 - (-3)}{1 - (-2)} = \frac{9 + 3}{1 + 2} = \frac{12}{3} = 4
\][/tex]
2. Find the y-intercept [tex]\( c \)[/tex]:
Using the point-slope form of the line [tex]\( y = mx + c \)[/tex]:
Substitute point [tex]\( A(-2, -3) \)[/tex] and the slope [tex]\( m = 4 \)[/tex]:
[tex]\[
-3 = 4(-2) + c \implies -3 = -8 + c \implies c = -3 + 8 = 5
\][/tex]
3. Equation of the line:
Substitute [tex]\( m = 4 \)[/tex] and [tex]\( c = 5 \)[/tex] into the line equation:
[tex]\[
y = 4x + 5
\][/tex]
So, the equation of the line [tex]\( AC \)[/tex] is:
[tex]\[
y = 4x + 5
\][/tex]
### (b) Calculate the acute angle between [tex]\( AC \)[/tex] and the [tex]\( x \)[/tex]-axis
To find the acute angle between the line [tex]\( AC \)[/tex] and the [tex]\( x \)[/tex]-axis, we use the fact that the tangent of the angle [tex]\( \theta \)[/tex] formed with the [tex]\( x \)[/tex]-axis is equal to the slope [tex]\( m \)[/tex]:
[tex]\[
m = \tan(\theta)
\][/tex]
Given the slope [tex]\( m = 4 \)[/tex]:
[tex]\[
\theta = \tan^{-1}(4)
\][/tex]
To express [tex]\( \theta \)[/tex] in degrees, we convert it from radians to degrees:
[tex]\[
\theta \approx 75.96^\circ
\][/tex]
So, the acute angle between the line [tex]\( AC \)[/tex] and the [tex]\( x \)[/tex]-axis is:
[tex]\[
75.96^\circ
\][/tex]
### (c) [tex]\( ABCD \)[/tex] is a kite, with [tex]\( AC \)[/tex] as the longer diagonal
Without additional specific information about points [tex]\( B \)[/tex] and [tex]\( D \)[/tex] or further details regarding the kite's dimensions and properties, it is not possible to provide a conclusive solution for this part. Typically, a kite has two pairs of adjacent sides that are equal in length, and its diagonals intersect at right angles. With such a geometric figure, knowing the exact coordinates or lengths is essential to proceed with further calculations or conclusions.
However, if the problem or additional context about points [tex]\( B \)[/tex] and [tex]\( D \)[/tex] is provided, the properties and relationships within the kite could be explored further.
