Theory question 112
Mensuration of Plane Shapes(a) In the diagram, AB is a tangent to the circle with centre O, and COB is a straight line. If CD ∥ AB and ∠ABE = 40°, find: ∠ODE.
(b) ABCD is a parallelogram in which |CD| = 7 cm, |AD| = 5 cm and ∠ADC = 125°. (i) Illustrate the information in a diagram. (ii) Find, correct to one decimal place, the area of the parallelogram.
(c) If x = ½(1 − √2), evaluate (2x2 − 2x).
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(a) ∠DCO = ∠ABO = 40° (alternate angles, CD ∥ AB)
OC = OD, so ∠ODC = 40° and ∠COD = 100°; ∠DOE = 180° − 100° = 80°
OD = OE, so ∠ODE = (180° − 80°)/2 = 50°
(b)(i) Sketch parallelogram ABCD with CD = 7 cm, AD = 5 cm, ∠ADC = 125°
(ii) Area = 7 × 5 × sin 125° ≈ 28.7 cm2
(c) 2x2 − 2x = 2x(x − 1); x − 1 = −½(1 + √2)
x(x − 1) = −¼(1 − √2)(1 + √2) = −¼(1 − 2) = ¼
2x2 − 2x = ½
In (c), factorising first makes the surds cancel neatly.