In a triangle length of two larger sides are 10 cm and 9 cm respectively . If the angles of the triangle are in A.P then the length of the third side can be?

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Answer 1 · 2017-09-21T17:53:25

# 7.449"cm, or, "2.551"cm."#

We will follow the Usual Notation for a #Delta.#

Given that, the angles of a triangle are in A.P.

So, we suppose that, in

#DeltaABC, A=x-y, B=x, &, C=x+y.#

But,

#A+B+C=180^@ :. (x-y)+x+(x+y)=180^@.#

# :. x=60^@.#

Since, #C>B>A, c>b>a.#

#:. c=10, b=9.#

Applying the Sine-Rule for #DeltaABC,# we have,

#a/sinA=b/sinB=c/sinC.#

#:. a/sin(60-y)=9/sin60^@=10/sin(60+y)...[because, B=x=60].#

#:. sin(60+y)=10/9*sin60~~0.96225.#

#:. 60+y=74.21^@, or, 180^@-74.21^@=105.79^@.#

#:. y=14.21^@, or, 45.79^@.#

#:. A=60-y=45.79^@, or, 14.21^@.#

#:. a/sin(45.79^@)=9/sin60^@, or, a/sin(14.21^@)=9/sin60^@.#

#:. a=(9sin45.79^@)/sin60^@, or, a=(9sin14.21^@)/sin60^@.#

#:. a~~7.449"cm, or, "a~~2.551"cm."#