Question

∆ RST is an equilateral triangle of side 9 cm. If O, Mand N are points on side RS, RT and ST respectivelysuch that OR - 3 cm and ROM = NMO = 90°Find the length of ON.​

Answer 1

Length of ON = 6 cm where ∆ RST is an equilateral triangle of side 9 cm. M and N are points on side RS, RT and ST respectively such that OR = 3 cm and ∠ROM = ∠NMO = 90°

Step-by-step explanation:

∆ RST is an equilateral triangle

=> ∠R = ∠S = ∠T = 60°

RS = ST = RT = 9 cm

in ΔROM

∠ROM = 90° ∠R = 60°  

=> ∠RMO = 30°  ( 180° - 90° - 60°)

Sin30 = RO/RM

=> 1/2 =  3 /RM

=> RM = 6 cm

OM² = RM² - RO² = 6² - 3²  = 27

MT = RT - RM = 9 - 6 = 3 cm

∠RMO = 30°

∠NMO = 90°

∠RMO + ∠NMO  + ∠NMT = 180°

=> ∠NMT = 180° - 30° - 90°

=> ∠NMT =  60°

=> Δ NMT is Equialteral triangle as ∠ T = 60° also

=> NM = MT

=> NM = 3 cm

in ΔNMO

ON² = OM² + NM²

=> ON² = 27 + 3²

=> ON² = 36

=> ON = 6 cm

Length of ON = 6 cm

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