In ∆ABC, If r1=8, r2=12, r3=24,Show that a=12,b=16,c=20. (From Properties of Triangles)
Answers
Step-by-step explanation:
Given, r₁ = 8, r₂ = 12, r₃ = 24.
∴ Derivative of (1/r) = (1/r₁) + (1/r₂) + (1/r₃)
⇒ (1/r) = (1/8) + (1/12) + (1/24)
⇒ (1/r) = (3 + 2 + 1)/24
⇒ r = 4.
Now,
∴ Δ = √rr₁r₂r₃
⇒ Δ = √4 * 8 * 12 * 24
= 96
(i)
r = Δ/s
⇒ 4 = 96/s
⇒ 4s = 96
⇒ s = 24.
(ii)
r₁ = Δ/s - a
⇒ 8 = 96/24 - a
⇒ 192 - 8a = 96
⇒ a = 12
(iii)
r₂ = Δ/s - b
⇒ 12 = 96/24 - b
⇒ 288 - 12b = 96
⇒ b = 16
(iv)
r₃ = Δ/s - c
⇒ 24 = 96/24 - c
⇒ 576 - 24c = 96
⇒ 480 = 24c
⇒ c = 20.
Therefore, a = 12, b = 16, c = 20.
Hope it helps!
Answer:
Step-by-step explanation:
Given lengths represent 'Pythagorean triplet' (18²+24²)=(30)²
a = 18, b = 24, c = 30 gives C = 90 degrees as a^2 + b^2 = c^2 then
cos A = b/c = 24/30 = 4/5 ANSWER
verify trigonometrically
cos A = (b^2 + c^2 -- a^2) / 2bc = (24^2 + 30^2 -- 18^2) / 2*24*30 = 1152/1440 = 4/5
cos B = a/c = 18/30 = 3/5 ANSWER
cos C = cos (90) = 0 ANSWER
sin A = a/c = 18/30 = 3/5 ANSWER
sin B = b/c = 24/30 = 4/5 ANSWER