ABCD i 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.s 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.
set up your vectors and it should be clear what's what.
Where do you get stuck?
To find the directed line segments AB, AD, BD, and AC in terms of a and b, we need to analyze the given information about ABCD being a square and P and Q being the midpoints of BC and CD, respectively.
Let's begin by visualizing the square ABCD:
A ------- B
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D ------- C
(i) AB:
Since ABCD is a square, opposite sides are parallel and equal in length. Thus, AB is equal to CD.
Considering that P is the midpoint of BC, we can say that AP = PC. Therefore, AP + PC = AC. Since AP is given as a, PC would also be a. Hence, AC = AP + PC = a + a = 2a.
(ii) AD:
Being a square, opposite sides of ABCD are parallel and equal in length. Consequently, AD is equal to BC.
Using the information that Q is the midpoint of CD, we can conclude that AQ = QD. So, AQ + QD = AD. Given that AQ is b, QD would also be b. Therefore, AD = AQ + QD = b + b = 2b.
(iii) BD:
As ABCD is a square, diagonals are perpendicular bisectors of each other. Consequently, BD is the diagonal linking B and D. Thus, BD = AC = 2a.
(iv) AC:
By using the information we gathered before, AC = 2a.
To sum up:
(i) AB = CD = AC = 2a
(ii) AD = BC = 2b
(iii) BD = 2a
(iv) AC = 2a
Hence, the directed line segments in terms of a and b are:
(i) AB = CD = AC = 2a
(ii) AD = BC = 2b
(iii) BD = 2a
(iv) AC = 2a