Question

1. find the missing variables.
2. find the missing variables.

Explanation:

Step1: Identify the triangle type

This is a \(45^{\circ}-45^{\circ}-90^{\circ}\) triangle. In a \(45^{\circ}-45^{\circ}-90^{\circ}\) triangle, the legs are equal (\(x = y\)), and the hypotenuse \(c\) is related to the leg \(a\) by the formula \(c=a\sqrt{2}\).

Step2: Solve for \(x\) and \(y\) in the first triangle

Given \(c = 2\sqrt{14}\). Using \(c=a\sqrt{2}\), we have \(2\sqrt{14}=x\sqrt{2}\).
Solve for \(x\):

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Since \(x = y\), \(y = 2\sqrt{7}\)

Step3: Solve for \(x\) and \(y\) in the second triangle

Given one leg \(a=\sqrt{10}\). Using \(c=a\sqrt{2}\), \(y=\sqrt{10}\times\sqrt{2}=\sqrt{20} = 2\sqrt{5}\). And since \(x\) is the other leg (in a \(45 - 45-90\) triangle legs are equal), \(x=\sqrt{10}\)

Answer:

For problem 6: \(x = 2\sqrt{7}\), \(y = 2\sqrt{7}\)
For problem 7: \(x=\sqrt{10}\), \(y = 2\sqrt{5}\)