A triangle is formed by the x-axis and the lines 5y = 9x and 5y + 9x = 45.
a Find (correct to the nearest degree) the angles of inclination of the two lines.
b What sort of triangle has been formed?
a) To find the angles of inclination of the two lines, we need to find the slopes of the lines. The slope of a line is given by the coefficient of x when the equation of the line is in the form y = mx + c, where m is the slope.
For the line 5y = 9x, we can rewrite it as y = (9/5)x. So the slope of this line is 9/5.
For the line 5y + 9x = 45, we can rewrite it as y = (-9/5)x + 9. So the slope of this line is -9/5.
The angle of inclination θ of a line with slope m is given by tan(θ) = m. Therefore, the angle of inclination for the line 5y = 9x is tan^(-1)(9/5) ≈ 59 degrees, and for the line 5y + 9x = 45, it is tan^(-1)(-9/5) ≈ -59 degrees.
b) Since the angles of inclination of the two lines are not equal, the triangle formed by the x-axis and the lines is a scalene triangle. This means that all three sides of the triangle are of different lengths, and all three angles are different.