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.

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