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?

Question

3132 views around the world

You can reuse this answer
Creative Commons License

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."#