Question

Determine a medida de x na figura a seguir:

Answer 1

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1ª Forma de Resolver :

Chamando a altura de y , temos :

$\Delta_{\text{maior}} \to (5+\text x)^2+\text y^2= (4\sqrt{5})^2$

$\Delta_{\text{menor}} \to \text x^2+\text y^2 = 5^2$

Subtraindo a 2ª da 1ª :

$(5+\text x)^2+\text y^2 - \text x^2 - \text y^2 = 80-25$

$25+10\text x + \text x^2 - \text x^2 =55$

$10\text x = 55 - 25$

$10\text x = 30$

$\huge\boxed{\text x = 3}\checkmark$

2ª Forma de resolver :

$\displaystyle \Delta_{\text {menor}} \to \text{Cos}(\theta) = \frac{\text x}{5}$

$\Delta_{\text{maior}} \to \text{Lei dos Cossenos } :\\\\\text{AC}^2 = \text{CD}^2+\text{AD}^2-2.\text{CD.AD}.\text{Cos}(180^{\circ}-\theta)$

$(4\sqrt5)^2=5^2+5^2-2.5.5.(-\text{Cos}(\theta))$

$\displaystyle 80=50+2.5.5.\frac{\text x}{5}$

$\displaystyle 10\text x = 30$

$\huge\boxed{\text x = 3}\checkmark$