ABCD is a square and P, Q are the midpoints of BC, CD respectively. If AP = a and AQ = b, find in terms of a and b, the directed line segments (i) AB, (ii) AD, (iii) BD and (iv) AC.
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Expert's answer
2021-01-25T09:53:30-0500
To find AB and AD we have to make some equations. First let us sign BP = x, AD = 2x, QD = y, AB = 2y. From the triangle ABP and from triangle AQD we can write two equations using the Pythagoras Theorem:
4y2+x2=a2
4x2+y2=b2
x2=a2−4y2
4(a2−4y2)+y2=b2
4a2−16y2+y2=b2
4a2−15y2=b2
y2=(4a2−b2)/15
y=(4a2−b2)/15;2y=2(4a2−b2)/15
x2=a2−4(4a2−b2)/15
x2=(15a2−16a2+4b2)/15
x2=(4b2−a2)/15
x=(4b2−a2)/15;2x=2(4b2−a2)/15
So AB=2y=2(4a2−b2)/15
AD=2x=2(4b2−a2)/15
From the triangle ABD we can find BD using the Pythagoras Theorem:
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