Question

(b) let x be the length of the hypotenuse. use the triangle you drew and the pythagorean theorem to fill in the blanks.
(c) find x, the length of the hypotenuse. give an exact answer (not a decimal approximation).
(d) each unit on the map represents 0.1 kilometers in real life. use your answer from part (c) to find the distance between the lighthouse and the campsite in real life. round your answer to the nearest tenth.

Explanation:

Step1: Recall Pythagorean Theorem

For a right - triangle with legs \(a\) and \(b\) and hypotenuse \(x\), the Pythagorean Theorem is \(a^{2}+b^{2}=x^{2}\). Since the problem is incomplete and we don't know the lengths of the legs, assume the legs have lengths \(a\) and \(b\). But if we assume the legs are of lengths \(6\) and \(x\) (from the given interface, though it's a bit unclear), the equation is \(6^{2}+x^{2}=x^{2}\) which is incorrect. Let's assume the legs are \(a\) and \(b\) and rewrite the Pythagorean equation as \(a^{2}+b^{2}=x^{2}\).

Step2: Solve for \(x\) in part (c)

If we assume the legs have lengths \(a\) and \(b\), then \(x = \sqrt{a^{2}+b^{2}}\). Since no values for \(a\) and \(b\) are given, we can't calculate a specific value. But if we assume the legs are \(6\) and some other value \(y\), then \(x=\sqrt{6^{2}+y^{2}}=\sqrt{36 + y^{2}}\).

Step3: Convert to real - life distance in part (d)

If \(x\) is the number of units on the map and each unit represents \(0.1\) kilometers in real - life, then the real - life distance \(d=0.1x\). Since we don't have a value for \(x\) from part (c), we can't calculate a real - life distance.

Since the problem is incomplete (no values for the legs of the right - triangle are given), we can't provide a final answer. But if we assume the legs of the right - triangle are \(a\) and \(b\), the general steps are as above.

If we assume the legs are \(6\) and \(8\) (for the sake of showing a complete example):

Step1: Apply Pythagorean Theorem

Let \(a = 6\) and \(b = 8\). Then \(a^{2}+b^{2}=x^{2}\), so \(6^{2}+8^{2}=x^{2}\), which is \(36 + 64=x^{2}\), and \(x^{2}=100\).

Step2: Solve for \(x\)

Taking the square root of both sides, \(x=\sqrt{100}=10\).

Step3: Convert to real - life distance

Since each unit represents \(0.1\) kilometers, the real - life distance \(d = 0.1\times10=1\) kilometer.

Answer:

Without given values for the legs of the right - triangle in the problem, we can't provide a definite answer. If the legs are \(6\) and \(8\), then:
For part (c): \(x = 10\)
For part (d): \(1\) kilometer