Question

In figure,AC=8cm,angleABC=90°,angle BAC= 60°, angle ACB =30° then find AB and BC

Answer 1

∠ABC = 94° and ∠BAC = 43°.

To find : ∠ABC and ∠BAC.

Given :

In ΔABC,

AB = BC

Here, AB = BC

Reason : When two sides of a triangles are equal then it is said to be an isosceles triangle.

Hence, ∠ACB = ∠BAC

In isosceles triangle, if two sides are equal then its opposite angles are equal

So, ∠BAC = 43°    [ ∵ ∠ACB = 43° ]

In triangle, sum of the angle is 180°.

∠ACB + ∠BAC + ∠ABC = 180°

43° + 43° + ∠ABC = 180°

        86° + ∠ABC = 180°

                 ∠ABC = 180° - 86°      

                 ∠ABC = 94°

Therefore, the value of ∠ABC is 94° and ∠BAC is 43°.

Answer 2

Given:

ΔABC

AC = 8cm

∠ABC = 90°

∠BAC = 60°

∠ACB = 30°

To find:

Length of sides AB and BC.

Solution:

From the figure, it can be seen that ∠ABC = 90°. That means ΔABC is a right triangle. Here, the length of the hypotenuse is given as $8cm$.

$Sin 30=\frac{OppositeSide}{Hypotenuse}$

$Sin30=\frac{AB}{AC}$

$\frac{1}{2}=\frac{AB}{8}$

$AB=\frac{8}{2}$

∴ $AB=4cm$

$Cos30=\frac{AdjacentSide}{Hypotenuse}$

$Cos30=\frac{BC}{8}$

$\frac{\sqrt{3} }{2} =\frac{BC}{8}$

$BC=\frac{8\sqrt{3} }{2}$

∴ $BC=4\sqrt{3}cm$

Length of side AB = 4cm

Length of side BC = $4\sqrt{3}$ cm