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Question
Mathematics
  1. Find the value of x x in the diagram below.
    (a) 8
    (b) 4
    (c) 3
    (d) 2
    (e) 1
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The answer is (d) 2.
Solution:
Using the angle sum property of a triangle, we have:
A + B + C = 180 2 x + 3 x + 4 x = 180 9 x = 180 x = 20 {:[/_A+/_B+/_C=180^(@)],[],[2x+3x+4x=180^(@)],[],[9x=180^(@)],[],[x=20^(@)]:}
Now, using the exterior angle theorem for triangles, we have:
D = A + B D = 2 x + 3 x D = 5 x D = 5 ( 20 ) D = 100 {:[/_D=/_A+/_B],[],[/_D=2x+3x],[],[/_D=5x],[],[/_D=5(20^(@))],[],[/_D=100^(@)]:}
Since the angles in a quadrilateral add up to 360 360^(@), using the angle sum property of quadrilaterals, we have:
C + D + E + F = 360 4 x + D + 5 x = 360 9 x + D = 360 180 + D = 360 D = 180 100 D = 80 {:[/_C+/_D+/_E+/_F=360^(@)],[],[4x+/_D+5x=360^(@)],[],[9x+/_D=360^(@)],[],[180^(@)+/_D=360^(@)],[],[/_D=180^(@)-100^(@)],[],[/_D=80^(@)]:}
Therefore, D E F = A B D + B C D = 2 x + 4 x = 6 x = 6 ( 20 ) = 120 . /_DEF=/_ABD+/_BCD=2x+4x=6x=6(20^(@))=120^(@).
Finally, we have:
E = 180 ( D + D E F ) = 180 ( 80 + 120 ) = 20 . {:[/_E=180^(@)-(/_D+/_DEF)],[],[=180^(@)-(80^(@)+120^(@))],[],[=-20^(@).]:}
Since E /_E cannot be negative, this is not a valid solution. Therefore, the value of x x is 2, and the answer is (d) 2.
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