Theory question 109
Constructions(a) Using a ruler and a pair of compasses only: (i) construct ΔXYZ such that |XY| = 8 cm and ∠YXZ = ∠ZYX = 45°; (ii) locate a point P inside the triangle equidistant from XY and XZ, and also equidistant from YX and YZ; (iii) construct a circle touching the three sides of the triangle. (iv) measure the radius of the circle.
(b) The length of the sides of a hexagon are x − 5, 2x, 2x + 7, 2x and 2x − 1. If the perimeter is 144 cm, find the value of x.
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(a)(i) Draw XY = 8 cm; construct 45° at X and at Y (bisect a constructed 90°); the arms meet at Z.
(ii) Bisect ∠X and ∠Y; P is where the bisectors meet.
(iii) Drop a perpendicular from P to XY; with centre P and that radius draw the inscribed circle.
(iv) Radius ≈ 1.7 cm (exact value 4√2 − 4 ≈ 1.66 cm).
(b) Perimeter: (x − 5) + 2x + 2x + (2x + 7) + 2x + (2x − 1) = 11x + 1 = 144, so x = 13.
The point equidistant from two lines lies on the bisector of the angle between them.