Question

11. (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: (i) number of sides; (ii) value of the exterior angle; (iii) sum of the interior angles of the polygon.

Accepted Answer

(a) Solve x+2y=7 and x−y=4 simultaneously: from the second equation, x=y+4. Substituting: (y+4)+2y=7 → 3y+4=7 → 3y=3 → y=1, x=5.
Intersection point = (5,1). Line through origin (0,0) and (5,1) has gradient = 1/5.
Equation: y = (1/5)x, i.e. x−5y=0.

(b) Interior:Exterior = 4:1. Since interior+exterior=180° (angles on a straight line): let exterior=e, interior=4e. 4e+e=180° → 5e=180° → e=36°.
(i) Number of sides = 360°/e = 360/36 = 10 sides.
(ii) Exterior angle = 36°.
(iii) Sum of interior angles = (n−2)×180° = (10−2)×180° = 8×180° = 1440°.

GCE Mathematics 2024 Theory — Question 11

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