Pythagoras’ Theorem：練習問題

Determine which of the following sets of numbers can represent the lengths of the sides of a right-angled triangle.

A point \(A\) has coordinates \((-5, 2)\) and point \(B\) has coordinates \((7, -3)\). What is the distance between point \(A\) and point \(B\)?

A small rectangular gift box has dimensions \( 2 \text{ cm} \) by \( 3 \text{ cm} \) by \( 6 \text{ cm} \). What is the distance between the two farthest points inside the box?

In a right-angled triangle, the lengths of the two legs are \( 10 \text{ cm} \) and \( 24 \text{ cm} \). What is the length of the hypotenuse?

In the figure, \(ABCD\) is a rectangle with \(AB = 12\text{ cm}\). If the length of the diagonal \(AC\) is \(20\text{ cm}\), find the area of the rectangle.

Find the length of the hypotenuse of a right-angled triangle whose legs are \( 12 \text{ cm} \) and \( 35 \text{ cm} \) long.

A hiker travels \(12 \text{ km}\) due North and then \(9 \text{ km}\) due East. From that position, she travels another \(8 \text{ km}\) due North and \(12 \text{ km}\) due East. Calculate the straight-line distance, in \(\text{km}\), from her starting point to her final destination.

In the figure, \(ABCD\) is a rectangle with \(AB = 24\text{ cm}\) and \(BC = 18\text{ cm}\). A point \(E\) lies on the diagonal \(AC\) such that \(AE:EC = 2:1\).
(a) Calculate the length of the diagonal \(AC\).
(b) Find the length of the segment \(BE\) correct to 1 decimal place.

A rectangular swimming pool is 15 meters long and 8 meters wide. A swimmer decides to swim diagonally from one corner to the opposite corner. How far does the swimmer travel in one diagonal lap?

A triangular garden \( ABC \) has side lengths \( AB = 9 \text{ m} \), \( BC = 12 \text{ m} \), and \( AC = 15 \text{ m} \).

(a) Using the converse of Pythagoras’ Theorem, show that \( \triangle ABC \) is a right-angled triangle. Which angle is the right angle?

(b) Calculate the area of the triangular garden.

(c) If a straight path is to be built from vertex \( B \) to the side \( AC \) such that the path is perpendicular to \( AC \), find the length of this path.