5.01 Pythagoras' Theorem

Name the hypotenuse in each triangle:

a

b

c

Use Pythagoras' theorem to determine whether the following triangles are right-angled:

a

b

c

d

Calculate the value of c in the following triangles:

a

b

c

d

e

f

g

Find the length of the hypotenuse of a right-angled triangle, to two decimal places, given that the other two sides are:

Calculate the value of a in the following triangles:

a

b

c

d

e

Calculate the value of b in the following triangles:

a

b

c

d

e

f

g

Find the value of b, to two decimal places, where b is the length of one of the shorter sides of a right-angled triangle which has:

Find the value of the pronumerals in the following diagrams. Round your answers to one decimal place.

a

b

c

d

The screen on a handheld device has dimensions 9 \text{ cm} by 5 \text{ cm}, and a diagonal of length x\text{ cm} . Find the value of x. Round your answer to two decimal places.

William and Kenneth are playing football together. At one point in the game, they are near the same corner of the field. William is on the goal line, 11 \text{ m} away from the corner, while Kenneth is on the side line, 17 \text{ m} away from the corner.

Find the shortest distance between William and Kenneth. Round your answer to two decimal places.

A movie director wants to shoot a scene where the hero of the film fires a grappling hook from the roof of one building to the roof of another. The shorter building is 37 \text{ m} tall, the taller building is 54 \text{ m} tall and the street between them is 10 \text{ m} wide.

Find the minimum length of rope, l, needed for the grappling hook. Give your answer correct to two decimal places.

A city council plans to build a seawall and boardwalk along a local coastline. According to safety regulations, the seawall needs to be 5.25 \text{ m} high and 7.66 \text{ m} deep and will be built at the bottom of a 14.78 \text{ m} long sloped section of shoreline. This means that the boardwalk will need to be built 2.43 \text{ m} above the seawall, so that it is level with the public area near the beach. This information is shown in the diagram below:

Find the width of the boardwalk, x \text{ m}, correct to two decimal places.

Consider the crane shown:

To help bear heavier loads, a support cable joins the end of one arm of the crane to the other, through a small tower that rises h \text{ m} above the crane arm.

Find, to two decimal places:

a

The value of h.

b

The value of l.

c

The total length of the support cable.

Quentin leaves his house in the morning and jogs 4.2 \text{ km} due north. He stops for lunch and then walks 2.8 \text{ km} due west to visit his sister, before jogging directly back to his house.

Find the total length of Quentin's afternoon journey, to one decimal place.

Barry wants to measure the length of his ladder, but he doesn't have a long enough tape measure. Instead, he leans the ladder up against a wall of vertical height 2.4 \text{ m.} He measures that the foot of the ladder rests at a horizontal distance of 1.8 \text{ m} away from the base wall.

Find the length of the ladder, to the nearest metre.

Neville is making a rectangular shaped pencil case. He decides to make one side of the pencil case 18 \text{ cm} wide, in order to fit his pens, and determines that the diagonal will need to be 20 \text{ cm} long, in order to fit his ruler.

Find the length of the other side of the pencil case. Round your answer to one decimal place.

A square prism has dimensions of 12 \text{ cm} by 12 \text{ cm} by 15 \text{ cm} as shown:

a

Calculate the length of HF, correct to two decimal places.

b

Calculate the length of DF, correct to two decimal places.

Consider the triangular prism below:

Find, to two decimal places:

a

The value of x.

b

The value of y.

c

The value of z.

Consider the following rectangular prism:

a

Find the value of x to two decimal places.

b

Find the value of y to two decimal places.

A new diagonal has been added as shown, with length z \text{ cm}:

c

Find the value of z to two decimal places.

d

Hence, find the area of the triangle bounded by the lengths a, y \text{ and } z.