Question

2 - dining room
find each missing side. then find the corresponding letters for each answer.

1. ac =
2. ab =
3. de =
4. df =

your lock code is your list of letters in order. enter your lock code below using all capitals and no spaces. example: rrqo
rrqo =
oops! try again!

Explanation:

Step1: Analyze right - triangle ABC

In right - triangle \(ABC\) with \(\angle B = 90^{\circ}\) and \(\angle C=45^{\circ}\), it is a \(45 - 45-90\) triangle. In a \(45 - 45 - 90\) triangle, the legs are equal and the hypotenuse \(c\) and legs \(a\) and \(b\) satisfy the relationship \(c = a\sqrt{2}\) (where \(a = b\)). Given \(BC = 4\), then \(AB=4\) (since in a \(45 - 45 - 90\) triangle \(AB = BC\)), and \(AC=\sqrt{AB^{2}+BC^{2}}=\sqrt{4^{2}+4^{2}}=\sqrt{16 + 16}=\sqrt{32}=4\sqrt{2}\).

Step2: Analyze right - triangle DEF

In right - triangle \(DEF\) with a right - angle at \(D\) and one leg \(DE\) and hypotenuse \(EF = 6\). Since it is also a \(45 - 45-90\) triangle (isosceles right - triangle), if the legs are \(x\), then the hypotenuse \(h=x\sqrt{2}\). Given \(h = EF = 6\), then \(x=\frac{6}{\sqrt{2}}=\frac{6\sqrt{2}}{2}=3\sqrt{2}\). So \(DE = DF=3\sqrt{2}\).

Answer:

1. \(AC = 4\sqrt{2}\) (corresponds to O)
2. \(AB = 4\) (corresponds to M)
3. \(DE = 3\sqrt{2}\) (corresponds to N)
4. \(DF = 3\sqrt{2}\) (corresponds to N)

The lock code is OMNN