Question

1. x = ______ y = ____ 5. x = ____ y = ____ 6. x = ____ y = ____ z = ______

Explanation:

Step1: Solve for \(x\) and \(y\) in the \(30 - 60-90\) triangle

In a \(30 - 60 - 90\) triangle, the sides are in the ratio \(1:\sqrt{3}:2\). If the side opposite the \(30^{\circ}\) angle is \(7\) (shorter leg), then the side opposite the \(60^{\circ}\) angle (\(x\), longer leg) is \(7\sqrt{3}\), and the hypotenuse (\(y\)) is \(14\).

Step2: Solve for \(x\) and \(y\) in the \(45 - 45-90\) triangle

In a \(45 - 45 - 90\) triangle, the sides are in the ratio \(1:1:\sqrt{2}\). Given the hypotenuse is \(6\), then the legs (\(x\) and \(y\)) are \(\frac{6}{\sqrt{2}}=3\sqrt{2}\).

Step3: Solve for \(x\), \(y\), and \(z\) in the larger \(45 - 45-90\) triangle

In a \(45 - 45 - 90\) triangle, if the hypotenuse is \(24\sqrt{2}\), then the legs (\(x\) and \(y\)) are \(\frac{24\sqrt{2}}{\sqrt{2}} = 24\). Assuming \(z\) is related to another part (but if it's a similar \(45 - 45 - 90\) or other standard triangle relation, if it's a leg - hypotenuse relation again, but if we assume it's a continuation of the same triangle logic as above, perhaps a mis - label. But if we consider basic \(45 - 45 - 90\) for the given hypotenuse \(24\sqrt{2}\), legs \(x=y = 24\).

Answer:

1. \(x = 7\sqrt{3}\), \(y = 14\)
2. \(x = 3\sqrt{2}\), \(y = 3\sqrt{2}\)
3. \(x = 24\), \(y = 24\) (assuming standard \(45 - 45 - 90\) triangle relations for the given hypotenuse \(24\sqrt{2}\))