Question

In a triangle the length of the two larger sides are 15cm

and 13 cm the angles of triangle are in an A.P. the

length of the remaining side can be​

Answer 1

Given :----

• Two larger sides of triangle are 15cm and 13cm.
• angles are in AP.
• We have to Find third side ?

Explaination :-----

First of all this is not a right angle triangle and question is absolutely correct . Excellent Question . We can solve it by Sine rule or cosine rule to Find rest two angles, we cant just assume its a right angled triangle for our comfort .

Solution :-----

Let us suppose that , angles of ∆ are (a-d), a and (a+d) since these are in AP .

Now, we know that , sum of all angles of ∆ are 180°.

so,

→ (a-d) + a + (a+d) = 180°

→ 3a = 180°

→ a = 60° .

Now, since , C > B > A , c > b > a

so, c = 15cm , and b = 13 cm

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Now, Lets solve it by sine rule , to Find Exact value of angles , [ we cant take d = 30°] .

Sine rule says that ,

→ [ a/sinA = b/sinB = c/sinC ]

Putting values we get,

→ a/sin(60-d) = 13/sin60 = 15/sin(60+d)

Now, comparing , we get,

→ sin(60+d) = 15*sin60/13 = 15*√3/2/13 [ as sin60° = √3/2]

→ (60+d) = sin^(-1)(0.9926008)

→ (60+d) = 83.025°

→ d = 83.025° - 60° = 23.025°

Putting now,

A = 60° - 23.025° = 36.975°

so,

→ a/sin36.975° = 13/sin60° = 15/sin83.025°

Comparing any now we get,

→ a = 13*sin36.975°/sin60° = 9.02 cm .

Again comparing now,

→ a = 15*sin36.975°/sin83.025° = 9.08cm .....

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So, we can say that, Approx values of third smaller side can be either 9.02 cm or 9.08cm ....

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we can solve this by Cosine rule also ,,

After finding angle B = 60°

→ cos60° = 15²+x²-13²/2*15*x

1/2 = 225 + x² - 169/30x

→ 15x = 56-x²

→ x² - 15x + 56 = 0

→ x² - 7x - 8x + 56 = 0

→ x(x-7) -8(x-7) = 0

→ (x-7) (x-8) = 0

if x - 7 = 0

x = 7cm ,

and if x - 8 = 0 ,

→ x = 8 cm

so, sides can be 7cm and 8cm also....

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Question was easy, only calculation part was tough, as in sine rule we have to use sine inverse .

i hope you understand both methods now, try yourself once ... i told you both methods .. Maths is all about practice yourself , just take hint and than try yourself .

Best of luck .

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