Pythagoras’ theorem：练习题

A right-angled triangle has two shorter sides with lengths of 9 cm and 12 cm. Calculate the length of the hypotenuse.

Calculate the distance between the points \(P(2, -3)\) and \(Q(6, 0)\) on a Cartesian coordinate grid.

A rectangular box (cuboid) has a length of 12 cm, a width of 4 cm, and a height of 3 cm. Calculate the length of the internal diagonal that connects a bottom corner to the opposite top corner.

A rectangular gate is 3 metres wide and 4 metres high. Calculate the length of a diagonal wooden brace that needs to be attached from one corner to the opposite corner.

An isosceles triangle has two equal sides of length 10 cm and a base of length 12 cm. Calculate the vertical height of this triangle.

A right-angled triangle has a hypotenuse of length \(13\text{ cm}\) and one side of length \(5\text{ cm}\). Calculate the length of the other side.

In the diagram, \(AC\) is perpendicular to \(BD\). Given that \(AB = 15\text{ cm}\), \(BC = 9\text{ cm}\), and \(CD = 16\text{ cm}\), calculate the length of \(AD\).

In a circle with centre \( O \), two parallel chords \( AB \) and \( CD \) are on the same side of the centre. Chord \( AB \) has length 16 cm and is 6 cm from \( O \). Chord \( CD \) is 3 cm from \( O \). Calculate the length of chord \( CD \), giving your answer to 1 decimal place.

A ladder of length \(6.5\text{ m}\) rests against a vertical wall. The base of the ladder is placed on horizontal ground, \(2.5\text{ m}\) away from the wall.

(a) Calculate the height up the wall reached by the top of the ladder.

(b) The ladder slips down so that its top is now \(5.2\text{ m}\) above the ground. Calculate how far the base of the ladder is now from the wall.

The diagram shows a right-angled trapezium \(ABCD\) where angle \(A = 90^\circ\) and angle \(B = 90^\circ\). The side lengths are \(AB = 12\text{ cm}\), \(AD = 18\text{ cm}\), and \(BC = 9\text{ cm}\).

(a) By constructing a perpendicular from \(C\) to \(AD\), calculate the length of side \(CD\).

(b) Calculate the total perimeter of trapezium \(ABCD\).

(c) Calculate the area of trapezium \(ABCD\).