Answer:
NO = 10 and OP = 18
Step-by-step explanation:
Given: KM = 48 , LD = 16 , NO : OP = 5 : 9 and NOPS is rectangle.
To find: Value of NO and OP
Let the value of NO and OP = 5x and 9x
frst we prove ΔKLD is similar to ΔKON and ΔMLD is similar to ΔMPS
In ΔKLD and ΔKON
∠KDL = ∠KNO = 90° ( corresponding angles )
∠DKL = ∠NKO ( common Angle )
⇒ ΔKLD is similar to ΔKON by AA similarity rule.
⇒ ![\frac{KD}{KN}=\frac{LD}{ON}](https://tex.z-dn.net/?f=%5Cfrac%7BKD%7D%7BKN%7D%3D%5Cfrac%7BLD%7D%7BON%7D)
subtract 1 from both sides
⇒ ![\frac{KD}{KN}-1=\frac{LD}{ON}-1](https://tex.z-dn.net/?f=%5Cfrac%7BKD%7D%7BKN%7D-1%3D%5Cfrac%7BLD%7D%7BON%7D-1)
⇒ ![\frac{KD-KN}{KN}=\frac{LD-ON}{ON}](https://tex.z-dn.net/?f=%5Cfrac%7BKD-KN%7D%7BKN%7D%3D%5Cfrac%7BLD-ON%7D%7BON%7D)
by substituting value from figure,
⇒ ![\frac{ND}{KN}=\frac{16-5x}{5x}](https://tex.z-dn.net/?f=%5Cfrac%7BND%7D%7BKN%7D%3D%5Cfrac%7B16-5x%7D%7B5x%7D)
⇒
..................... (1)
In ΔMLD and ΔMPS
∠MDL = ∠MSP = 90° ( corresponding angles )
∠DML = ∠SMP ( common Angle )
⇒ ΔMLD is similar to ΔMPS by AA similarity rule.
⇒ ![\frac{MD}{SM}=\frac{LD}{PS}](https://tex.z-dn.net/?f=%5Cfrac%7BMD%7D%7BSM%7D%3D%5Cfrac%7BLD%7D%7BPS%7D)
subtract 1 from both sides
⇒ ![\frac{MD}{SM}-1=\frac{LD}{PS}-1](https://tex.z-dn.net/?f=%5Cfrac%7BMD%7D%7BSM%7D-1%3D%5Cfrac%7BLD%7D%7BPS%7D-1)
⇒ ![\frac{MD-SM}{SM}=\frac{LD-PS}{PS}](https://tex.z-dn.net/?f=%5Cfrac%7BMD-SM%7D%7BSM%7D%3D%5Cfrac%7BLD-PS%7D%7BPS%7D)
by sustituting value from figure,
⇒ ![\frac{DS}{SM}=\frac{16-5x}{5x}](https://tex.z-dn.net/?f=%5Cfrac%7BDS%7D%7BSM%7D%3D%5Cfrac%7B16-5x%7D%7B5x%7D)
⇒
..................... (2)
Add eqn. (1) & (2), we get
![ND+DS=(\frac{16-5x}{5x})\times KN + (\frac{16-5x}{5x})\times SM](https://tex.z-dn.net/?f=ND%2BDS%3D%28%5Cfrac%7B16-5x%7D%7B5x%7D%29%5Ctimes%20KN%20%2B%20%28%5Cfrac%7B16-5x%7D%7B5x%7D%29%5Ctimes%20SM)
![NS=(\frac{16-5x}{5x})(KN+SM)](https://tex.z-dn.net/?f=NS%3D%28%5Cfrac%7B16-5x%7D%7B5x%7D%29%28KN%2BSM%29)
( from figure KN + SM = KM - NS)
substitute given values,
![9x=(\frac{16-5x}{5x})(48-9x)](https://tex.z-dn.net/?f=9x%3D%28%5Cfrac%7B16-5x%7D%7B5x%7D%29%2848-9x%29)
![9x \times 5x=16\times48+16\times(-9x)-5x\times48-5x\times(-9x)](https://tex.z-dn.net/?f=9x%20%5Ctimes%205x%3D16%5Ctimes48%2B16%5Ctimes%28-9x%29-5x%5Ctimes48-5x%5Ctimes%28-9x%29)
![45x^2=768-144x-240x+45x^2](https://tex.z-dn.net/?f=45x%5E2%3D768-144x-240x%2B45x%5E2)
![45x^2-45x^2=768-384x](https://tex.z-dn.net/?f=45x%5E2-45x%5E2%3D768-384x)
![768-384x=0](https://tex.z-dn.net/?f=768-384x%3D0)
![384x=768](https://tex.z-dn.net/?f=384x%3D768)
![x=\frac{768}{384}](https://tex.z-dn.net/?f=x%3D%5Cfrac%7B768%7D%7B384%7D)
x = 2
⇒ NO = 5x = 5 × 2 = 10
⇒ OP = 9x = 9 × 2 = 18