Question

In the figure given below, 𝑃𝐴, 𝑄𝐵, 𝑅𝐶 and 𝑆𝐷 are all perpendiculars to a line 𝑙, 𝐴𝐵 = 6 𝑐𝑚, 𝐵𝐶 = 9 𝑐𝑚, 𝐶𝐷 = 12 𝑐𝑚 and 𝑆𝑃 = 36 𝑐𝑚. Find 𝑃𝑄, 𝑄𝑅 and 𝑅𝑆.

Answer 1

From the given data, we have 𝐴𝐵 = 6 𝑐𝑚, 𝐵𝐶 = 9 𝑐𝑚, 𝐶𝐷 = 12 𝑐𝑚 and 𝑆𝑃 = 36 𝑐𝑚.
And, PA, QB, RC and SD are perpendicular to the same line l.
Let us consider that line QB, RC cut line PD at X and Y respectively.

Now, in triangle PSD, we have
QX || RY || SD [∵ AP || QB || RC || SD]
∴ By BPT theorem, we have
PQ : QR : RS = PX : XY : YD ……(i)

Now, in triangle PDA, we have
AP || XB || YC
[∵ AP || QB || RC || SD]
∴ By BPT theorem, we have
PX : XY : YD = AB : BC : CD …..(ii)

From eq (i) and (ii) we have,
PQ : QR : RS = AB : BC : CD
PQ : QR : RS = 6 : 9 : 12

Let PQ be 6x , QR be 9x and RS be 12x also we have,
PS = PQ + QR + RS
36 = 6x + 9x + 12x
36 = 27x
x = 4/3

Now, putting the value of x = 4/3 in PQ , QR and RS we have,
PQ = 6x = 8cm
QR = 9x = 12 cm
RS = 12x = 16 cm