Question

right triangle trigonometry: advanced
google classroom microsoft teams
what is the value of ( x ) in the figure shown?
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how do i enter a student - produced response on the sat? show me!

Explanation:

Step1: Find the length of \(AB\)

In right - triangle \(ABC\) with \(\angle B = 30^{\circ}\) and \(\angle C = 90^{\circ}\), if the side opposite to \(30^{\circ}\) is \(AC = 4\), then the hypotenuse \(AB\) is \(8\) (since in a \(30 - 60-90\) triangle, the hypotenuse is twice the side opposite the \(30^{\circ}\) angle, \(AB = 2AC\)).

Step2: Find the length of \(AD\)

In right - triangle \(ACD\), using the Pythagorean theorem \(AD=\sqrt{AC^{2}-CD^{2}}\). But also, since \(\angle A = 60^{\circ}\) (because \(\angle A+\angle B = 90^{\circ}\) in \(\triangle ABC\)), and \(\cos A=\frac{AD}{AC}\), \(\cos60^{\circ}=\frac{AD}{4}\), so \(AD = 4\times\frac{1}{2}=2\).

Step3: Calculate \(x\)

Since \(x=AB - AD\), substituting \(AB = 8\) and \(AD = 2\), we get \(x=8 - 2\).

Answer:

\(6\)