Question

find the value of each variable. write your answer as an integer or in simplified radical form.
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7 home plate, first base, second base, and third base form a perfect square. the distance from each consecutive
is 95 feet. what is the distance from home plate directly to second base?

Explanation:

1.

Step1: Use properties of \(45 - 45-90\) triangle

In a \(45 - 45-90\) triangle, the legs are equal (\(x = 13\)) and the hypotenuse \(y=\sqrt{2}\times\) leg. So \(y = 13\sqrt{2}\).

2.

Step1: Use properties of \(45 - 45-90\) triangle

In a \(45 - 45-90\) triangle, the legs are equal (\(x=y\)) and the hypotenuse \(h=\sqrt{2}\times\) leg. Given \(h = 50\), then \(x=y=\frac{50}{\sqrt{2}}=\frac{50\sqrt{2}}{2}=25\sqrt{2}\).

3.

Step1: Use properties of \(45 - 45-90\) triangle

In a \(45 - 45-90\) triangle, the legs are equal (\(x = 7\)) and the hypotenuse \(y=\sqrt{2}\times\) leg. So \(y = 7\).

4.

Step1: Use properties of \(45 - 45-90\) triangle

In a \(45 - 45-90\) triangle, the legs are equal (\(x=\sqrt{10}\)) and the hypotenuse \(y=\sqrt{2}\times\) leg. So \(y=\sqrt{2}\times\sqrt{10}=\sqrt{20}=2\sqrt{5}\).

5.

Step1: Use properties of \(45 - 45-90\) triangle

In a \(45 - 45-90\) triangle, the legs \(x=\frac{24}{\sqrt{2}}=\frac{24\sqrt{2}}{2}=12\sqrt{2}\).

6.

Step1: Use properties of \(45 - 45-90\) triangle

In a \(45 - 45-90\) triangle, the legs are equal. Given one leg \(=6\), so \(x = 6\).

7.

Step1: Use properties of \(45 - 45-90\) triangle

The distance between consecutive bases is the leg (\(a = 90\) feet) of a \(45 - 45-90\) triangle. The distance from home - plate to second - base (hypotenuse \(d\)) is \(d=\sqrt{2}\times a\). So \(d = 90\sqrt{2}\) feet.

Answer:

1. \(x = 13,y = 13\sqrt{2}\)
2. \(x = 25\sqrt{2},y = 25\sqrt{2}\)
3. \(x = 7,y = 7\)
4. \(x=\sqrt{10},y = 2\sqrt{5}\)
5. \(x = 12\sqrt{2}\)
6. \(x = 6\)
7. \(90\sqrt{2}\) feet