(4) $$ \operatorname{In} \Delta X Y Z, X=y, \frac{z^{2}-2 x^{2}}{x^{2}}=1 $$, then $$ m(\angle Z)=\ldots $$ $$ \begin{array}{llll} ext { (a) } 30^{\circ} & ext { (b) } 60^{\circ} & ext { (c) } 120^{\circ} & ext { (d) } 150^{\circ}\end{array} $$

Question

(4) $$ \operatorname{In} \Delta X Y Z, X=y, \frac{z^{2}-2 x^{2}}{x^{2}}=1 $$, then $$ m(\angle Z)=\ldots $$ $$ \begin{array}{llll}	ext { (a) } 30^{\circ} & 	ext { (b) } 60^{\circ} & 	ext { (c) } 120^{\circ} & 	ext { (d) } 150^{\circ}\end{array} $$

UpStudy AI · Accepted Answer

The measure of angle $$Z$$ is $$120^\circ$$. To solve the problem, we start with the given equation: $$ \frac{z^2 - 2x^2}{x^2} = 1 $$ We can rearrange this equation: $$ z^2 - 2x^2 = x^2 $$ This simplifies to: $$ z^2 = 3x^2 $$ Taking the square root of both sides, we find: $$ z = \sqrt{3}x $$ Now we have the lengths of the sides of triangle $$XYZ$$ in terms of $$x$$: - $$XY = x$$ - $$XZ = z = \sqrt{3}x$$ - $$YZ = y = x$$ Next, we can use the Law of Cosines to find the angle $$Z$$: $$ c^2 = a^2 + b^2 - 2ab \cos(C) $$ In our case, let: - $$a = XY = x$$ - $$b = YZ = x$$ - $$c = XZ = \sqrt{3}x$$ Substituting these values into the Law of Cosines gives: $$ (\sqrt{3}x)^2 = x^2 + x^2 - 2(x)(x) \cos(Z) $$ This simplifies to: $$ 3x^2 = 2x^2 - 2x^2 \cos(Z) $$ Rearranging this, we have: $$ 3x^2 = 2x^2(1 - \cos(Z)) $$ Dividing both sides by $$x^2$$ (assuming $$x \neq 0$$): $$ 3 = 2(1 - \cos(Z)) $$ Solving for $$\cos(Z)$$: $$ 3 = 2 - 2\cos(Z) $$ $$ 2\cos(Z) = 2 - 3 $$ $$ 2\cos(Z) = -1 $$ $$ \cos(Z) = -\frac{1}{2} $$ The angle $$Z$$ for which $$\cos(Z) = -\frac{1}{2}$$ is: $$ Z = 120^\circ $$ Thus, the measure of angle $$Z$$ is: $$ \boxed{120^\circ} $$

(4) \( \operatorname{In} \Delta X Y Z, X=y, \frac{z^{2}-2 x^{2}}{x^{2}}=1 \), then \( m(\angle Z)=\ldots \) \( \begin{array}{llll}\text { (a) } 30^{\circ} & \text { (b) } 60^{\circ} & \text { (c) } 120^{\circ} & \text { (d) } 150^{\circ}\end{array} \)

UpStudy ThothAI Solution

Step-by-step Solution

Quick Answer

(4) \( \operatorname{In} \Delta X Y Z, X=y, \frac{z^{2}-2 x^{2}}{x^{2}}=1 \), then \( m(\angle Z)=\ldots \) \( \begin{array}{llll}\text { (a) } 30^{\circ} & \text { (b) } 60^{\circ} & \text { (c) } 120^{\circ} & \text { (d) } 150^{\circ}\end{array} \)

UpStudy ThothAI Solution

Step-by-step Solution

Quick Answer

Related Questions