(a) Find the equation of the line that passes through the origin and the point of intersection of the lines \( x + 2y = 7 \) and \( x – y = 4 \). (b) The ratio of an interior angle to an exterior angle of a regular polygon is 4: 1. Find the: (i) number of sides; (ii) value of the exterior angle; and (iii) sum of the interior angles of the polygon.
Explanation
(a) Solving the eqns of the two lines simultaneously:
x + 2y = 7 - - -- - - - - - -(i)
x - y = 4 - - - - - - - - - - - -(ii)
subtract eqn (ii) from (i)
x - x + 2y - (- y) = 7 - 4
3y = 3
y = 1
put y = 1 into eqn (ii)
x - 1 = 4
x = 4 + 1 = 5
The point of intersection of lines is (5, 1)
⇒ If the line passes through Origin (0, 0) and (5, 1)
Using: \(\frac{y_2 - y_1}{x_2 - x_1} = \frac{y - y_1}{x - x_1}\)
\(\frac{1 - 0}{5 - 0} = \frac{y - 0}{x - 0}\)
\(\frac{1}{5} = \frac{\text{y}}{\text{x}}\)
Therefore, the equation of the line = x - 5y = 0
(b) Given that the ratio of the interior angle to the exterior angle of a regular polygon = 4: 1
Thus, I = 4E
But, the sum of the interior angles and the exterior angles of a regular polygon = 180
i.e. I + E = 180
4E + E = 180
5E = 180
E = \(\frac{180}{5}\) = 36º
Since, I = 4E = 4 x 36 = 144º
(ii) value of the exterior angle = 36º
(i) Number of sides of the polygon = \(\frac{360}{\text{n}}\) = E, n = \(\frac{360}{36}\) = 10 (since E = 36º)
(iii) Sum of the interior angles of the polygon. = (n - 2)180 = (10 - 2)180 = 8 x 180 = 1,440.