Question

If the angles A,B,C of ∆ABC are in A.P and angleA = 30°, C=5 , then find values of a and b​

Answer 1

Given :

• Angles of ΔABC are in AP
• ∠A = 30° , c = 5.

To Find :

• The values of a and b

Solution :

The general condition when a , b , c are in AP is 2b = a + c.

Here , we are given that ∠A , ∠B and ∠C (three angles of ΔABC) are in AP.

$\\ \implies \rm\angle \: A + \angle \: C = 2( \angle \: B) \: \: ...(1)$

By angle sum property of a triangle ,

$\\ \implies \rm \angle \: A \: + \angle \: B + \angle \: C= {180}^{ \circ}$

From equation(1) ;

$\\ \implies \rm\angle \: 2B + \angle \: B= {180}^{ \circ}$

$\implies \rm \angle \: B = \dfrac{180}{3}$

$\\ \implies {\boxed{\rm{ \angle \: B = {60}^{ \circ} }}}$

Now we have ,

• ∠A = 30° , ∠B = 60°

Substituting these values in the equation(1) we get ;

$\\ \implies \rm \: {30}^{ \circ} + \angle \: C=2( 60 {}^{ \circ})$

$\\ \implies \rm \angle \: C = 120^{ \circ} - {30}^{ \circ}$

$\\ : \implies {\boxed{\rm{ \angle \: C= {90}^{ \circ} }}}$

Now we have ;

• ∠A = 30° , ∠B = 60° , ∠C = 90°.

Applying sine rule [properties of triangle] formula ,

$\\ {\boxed{\bf{ \dfrac{a}{ \sin(A)} = \dfrac{b}{ \sin(B) } = \dfrac{c}{ \sin(C)} }}}$

Here ,

• a , b , c are lengths of the sides of triangle. [ a is length of BC , b is length of AC and c is length of AB]
• Sin(A) , Sin(B) , Sin(C) are the angles.

We have the measure of ∠A of this triangle as 30°. So , Sin(A) for the given triangle is

$\\ \implies \: \rm \sin(A) = \sin( {30}^{ \circ} ) = \frac{1}{2}$

Similarly ,

$\\ \implies \rm \sin(B) = \sin( {60}^{ \circ} ) = \dfrac{ \sqrt{3} }{2}$

$\\ \implies \rm \: \sin(C) = \sin( {90}^{ \circ} ) = 1$

Considering this relation ,

$\\ \implies \rm \: \dfrac{a}{ \sin(A)} = {\underbrace{\rm{\dfrac{b}{ \sin(B) } = \dfrac{c}{\sin(C)}}}}$

$\\ \implies \rm \dfrac{b}{ \sin(B) } = \dfrac{c}{ \sin(C) }$

We have the values of c , Sin(C) and Sin(B). Substituting the values in the equation we get ;

$\\ \implies \rm \dfrac{b}{ \frac{ \sqrt{3} }{2} } = \dfrac{5}{1}$

$\\ \implies {\boxed{\rm{\blue{b = \dfrac{5 \sqrt{3} }{2} }}}}$

Now consider the relation ,

$\\ \implies {\underbrace{\rm{\dfrac{a}{ \sin(A)} = \dfrac{b}{ \sin(B) } }}}= \dfrac{c}{ \sin(C) }$

$\\ \implies \: \dfrac{a}{ \sin(A) } = \dfrac{b}{ \sin(B) }$

We have the values of Sin(A) , b and Sin(B). Substituting the values in the equation we get,

$\\ \implies \rm \dfrac{a}{ \frac{1}{2} } = \dfrac{5( \frac{ \sqrt{3} }{2}) }{ \frac{ \sqrt{3} }{2} }$

$\\ \implies \rm \: 2a = 5$

$\\ \implies{\boxed{\blue{\rm{a = \frac{5}{2} }}}}$

Hence ,

• The values of a and b are $\sf{\dfrac{5}{2}}$ and $\sf{\dfrac{5\sqrt{3}}{2}}$ respectively.

Answer 2

Step-by-step explanation:

(b) we have A+B+C= 180° .

Also A, B, C are in A. p = 2B = A+c.

:- 3B=180° or B=60°

now b² = a² + c² - 2ac cos B

= a²+c²-2ac cos 60°

or b²= a² + c² - ac.

i hope full help it.

b. option is correct✅