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The problem involves finding the shortest distance between two planets, \(Q\) and \(R\), using the given distances:

- The distance from Planet \(Q\) to Planet \(P\) is \(3.6 \times 10^6\) km.
- The distance from Planet \(R\) to Planet \(P\) is \(1.6 \times 10^7\) km.

From the diagram, it appears that the triangle formed by the three planets is a right-angled triangle, where:

- The distance between \(P\) and \(Q\) is one leg of the triangle.
- The distance between \(P\) and \(R\) is the other leg of the triangle.

We can use the Pythagorean theorem to find the shortest distance (the hypotenuse) between Planet \(Q\) and Planet \(R\). The formula for the Pythagorean theorem is:

\[
c = \sqrt{a^2 + b^2}
\]

Where:
- \(a = 3.6 \times 10^6\) km (distance between \(Q\) and \(P\)),
- \(b = 1.6 \times 10^7\) km (distance between \(R\) and \(P\)).

Let's calculate that now.The shortest distance between Planet \(Q\) and Planet \(R\) is \(1.64 \times 10^7\) km, in standard form.

Would you like more details or have any questions? Here are some related questions to further expand your understanding:

1. How is the Pythagorean theorem applied to different geometrical shapes?
2. What are the real-world applications of the Pythagorean theorem in astronomy?
3. How do you convert between standard form and scientific notation?
4. What other methods can be used to calculate distances in space?
5. How would rounding affect the precision of distances in scientific contexts?

**Tip:** In problems involving large distances like those in space, using scientific notation helps simplify calculations and avoid errors in handling very large numbers.

Work out the shortest distance between planet Q and planet R. Given the distances: Planet Q is 3.6 x 10^6 km from planet P, and Planet R is 1.6 x 10^7 km from planet P.

Learn how to find the shortest distance between two planets using the Pythagorean theorem. This example involves calculating the hypotenuse of a right-angled triangle formed by distances of 3.6 x 10^6 km and 1.6 x 10^7 km from a common point, ideal for high school students learning advanced geometry.

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