Question
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 Find (correct to the nearest degree) the angles of inclination of the two lines.
b What sort of triangle has been formed?
Answers
Answered by
GPT 3.5
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.
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.
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