From the given figure, by pythogorous theorem \(\mathrm{PR}^{2}=\mathrm{PQ}^{2}+\mathrm{R} \mathrm{Q}^{2}\) \(\mathrm{PR}^{2}=4^{2}+3^{2}\) \(\mathrm{PR}^{2}=16+9\) \(\mathrm{PR}=\sqrt{25}\) \(\mathrm{PR}=5 \mathrm{~m}\) Average speed \(=\frac{\text { total distance }}{\text { total time }}=\frac{4 \mathrm{~m}+3 \mathrm{~m}+5 \mathrm{~m}}{20 \mathrm{~s}+10 \mathrm{~s}+30 \mathrm{~s}}=\frac{12 \mathrm{~m}}{60 \mathrm{~s}}=0.2 \mathrm{~m} \mathrm{~s}^{-1}\)

One of the fundamental shapes in geometry that students often encounter is the right triangle. It is characterized by a 90-degree angle, making one corner exactly square. This triangle's sides have special names: the longest side, opposite the right angle, is known as the hypotenuse, while the other two sides are called the legs. To identify these, one just looks for the side that is alone on one side of the right angle, which is the hypotenuse, whereas the legs form the actual right angle.

When solving problems involving right triangles, the Pythagorean Theorem is an invaluable tool. It states that \(a^2 + b^2 = c^2\), where \(a\) and \(b\) represent the lengths of the legs, and \(c\) is the length of the hypotenuse. This theorem allows us to determine the length of one side when the lengths of the other two are known, as demonstrated in the provided example where the hypotenuse length was found using the lengths of the legs.

In the given exercise example, \( PR \) is the hypotenuse, and \( PQ \) and \( RQ \) are the legs forming the right angle. By applying the Pythagorean theorem, we quickly deduce the hypotenuse length, crucial for further calculations.

Average speed calculation is another crucial concept that goes beyond geometry, entering the realm of physics. To calculate the average speed, one must take the total distance traveled and divide it by the total time taken to cover that distance. The formula for average speed is: \(\text{average speed} = \frac{\text{total distance}}{\text{total time}}\).

Understanding this concept requires an awareness of consistent units (e.g., meters per second, \(\mathrm{m~s}^{-1}\)) and clarity that average speed is not necessarily the speed at a particular moment, but rather an overall measurement of a completed journey's speed. For instance, the exercise provided details distances covered and times taken for each leg of a journey. By adding these individual distances and times, we can calculate the object's average speed over the entire trip using the formula mentioned.

The last key point to understand from our exercise is how to determine the lengths of different sides of a triangle. In right triangles, as previously mentioned, the Pythagorean Theorem is essential. However, when dealing with any triangle, knowing all side lengths can be crucial for solving many types of geometric problems such as finding area, perimeter, or other sides using various rules including the sine and cosine rules.

In the case of the right triangle from our exercise, we found the hypotenuse length by calculating the square root of the sum of the squares of the other two sides. This step is based purely on the Pythagorean Theorem. It is critical, especially for students, to remember that it only applies to right triangles. Knowing the side lengths also paves the way to explore trigonometry, where we can find out the angles of a triangle based on side lengths, which is incredibly useful in many practical applications like engineering, physics, and even navigation.