Answer:
AB = 7 cm
Step-by-step explanation:
Sine Rule for Area
[tex]\sf Area =\dfrac{1}{2}ab \sin C[/tex]
where:
- a, b and c are the sides opposite angles A, B and C
- a and b are the sides and C is the included angle
Given:
- Area = √(300) cm²
- a = (x + 3) cm
- b = x cm
- C = 60°
Substitute the given values into the formula and solve for x:
[tex]\begin{aligned} \sf Area & = \dfrac{1}{2}ab \sin C \\\\\implies \sqrt{300} & = \dfrac{1}{2}(x+3)x \sin 60^{\circ}\\\\\sqrt{300} & = \dfrac{\sqrt{3}}{4}(x+3)x\\\\\dfrac{4\sqrt{300}}{\sqrt{3}} & = x^2+3x\\\\40 & = x^2+3x\\\\x^2+3x-40 & = 0\\\\ (x-5)(x+8)& = 0\\\\ \implies x & = 5, -8\end{aligned}[/tex]
As length is positive, x = 5 only.
Substitute the found value of x into the expressions for the side lengths:
Cosine rule
[tex]c^2=a^2+b^2-2ab \cos C[/tex]
(where a, b and c are the sides and C is the angle opposite side c)
Substitute the found values into the formula and solve for AB:
[tex]\begin{aligned}c^2 & =a^2+b^2-2ab \cos C\\\implies AB^2 & =8^2+5^2-2(8)(5) \cos 60^{\circ}\\AB^2 & =89-40\\AB^2 & =49\\AB & =\sqrt{49}\\AB & = \pm 7\end{aligned}[/tex]
As length is positive, AB = 7 cm.
Learn more about the sine rule for area here:
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