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#1**+2 **

So, \(\triangle RST\) has two given interior angles, 60^{o} and 90^{o}. Since the sum of the interior angles of a triangle is 180^{o}, we can deduce that the remaining angle has a measure of 30^{o}. This is a special triangle in which the hypotenuse is double the shorter leg, and the longer leg is \(\sqrt{3}\) times the shorter leg. Since \(\overline{RS}\) is the longer leg and has a measure of \(2\sqrt3\), the measure of \(\overline{ST}\) is \({2\sqrt3\over\sqrt3}=2\). This means that the measure of hypotenuse \(\overline{RT}\) is equal to \(2*2=4\). Now, \(\triangle QRT\) also has two given angles, 45^{o} and 90^{o}. By the same logic as before, the remaining angle has a measure of 45^{o}. This is also a special triangle, in which the two legs are equal, and the hypotenuse is \(\sqrt2\) times the leg. Since \(\overline{RT}\) and \(\overline{ RQ}\) are both legs of the triangle, their measures must be equal. Therefore, \(x=4\).

Mathhemathh Apr 3, 2018

#1**+2 **

Best Answer

So, \(\triangle RST\) has two given interior angles, 60^{o} and 90^{o}. Since the sum of the interior angles of a triangle is 180^{o}, we can deduce that the remaining angle has a measure of 30^{o}. This is a special triangle in which the hypotenuse is double the shorter leg, and the longer leg is \(\sqrt{3}\) times the shorter leg. Since \(\overline{RS}\) is the longer leg and has a measure of \(2\sqrt3\), the measure of \(\overline{ST}\) is \({2\sqrt3\over\sqrt3}=2\). This means that the measure of hypotenuse \(\overline{RT}\) is equal to \(2*2=4\). Now, \(\triangle QRT\) also has two given angles, 45^{o} and 90^{o}. By the same logic as before, the remaining angle has a measure of 45^{o}. This is also a special triangle, in which the two legs are equal, and the hypotenuse is \(\sqrt2\) times the leg. Since \(\overline{RT}\) and \(\overline{ RQ}\) are both legs of the triangle, their measures must be equal. Therefore, \(x=4\).

Mathhemathh Apr 3, 2018