9.04 Triples and 3D problems (Extended)
Determine whether the following are Pythagorean triples:
\left(5, 12, 13\right)
\left(12, 5, 13\right)
\left(8, 14, 18\right)
\left(9, 16, 25\right)
\left(300, 400, 500\right)
\left(9, 12, 15\right)
\left(15, 8, 17\right)
\left(\dfrac{1}{3}, \dfrac{1}{4}, \dfrac{1}{5}\right)
Complete the following Pythagorean triples:
⬚, 15, 17
7, ⬚, 25
28, ⬚, 53
14, 48, ⬚
Peter knows the two smallest numbers in a Pythagorean triple, which are 3 and 4. Find the number Peter needs to complete the triple.
Sean knows the two largest numbers in a Pythagorean triple, which are 41 and 40. Find the number Sean needs to complete the triple.
Given that the length of the hypotenuse of a right-angled triangle is 20, find the two other side lengths that would complete a Pythagorean triple.
Find the number that completes a Pythagorean triple given the following:
The two largest numbers are 100 and 96.
The two smallest numbers are 22 and 120.
If x, y, and z form a Pythagorean triple, will 3 x, 3 y and 3 z be a Pythagorean triple?
Determine whether the following triples of numbers represent the sides of right-angled triangles:
\left(2, 4, 6\right)
\left(5, 12, 13\right)
\left(6, 8, 13\right)
\left(3, 4, 5\right)
Identify the hypotenuse of the following triangles:
For each of the following triangles, let a and b represent the two shorter sides and c the length of the longest side:
Find a^{2} + b^{2}.
Find c^{2}.
Is the triangle a right-angled triangle?
Consider the following right-angled triangle:
Write the equation that can be constructed from the given information.
A right-angled triangle has side lengths x, y, and z. Will a triangle with side lengths of 5 x, 5 y and 5 z make a right-angled triangle as well?
Consider the right-angled triangle with sides \left(8, 15, 17\right).
Find the length of the side that is opposite the largest angle.
Find the lengths of the two sides that are next to the right angle.
Consider a right-angled triangle with shorter side lengths of 15 units and 36 units.
Find the length of the hypotenuse in the triangle.
The triple is a multiple of a common Pythagorean triple. Which one?
Consider a right-angled triangle with side length of 40 \text{ cm} and hypotenuse of 85 \text{ cm}.
Find the length of the unknown side in the triangle.
The triple is a multiple of a common Pythagorean triple. Which one?
A group of engineering students have made a triangle out of some wooden strips. They have made a triangle with sides lengths 20 \text{ m}, 48 \text{ m}, and 52 \text{ m}.
Is the triangle they make a right-angled triangle?
How many metres of wooden strips did they use to make the triangle?
Find the other right-angled triangles they could create using the exact same length of wooden strips.
All edges of the following cube are 5 cm long. Find, to two decimal places:
The value of x.
The value of y.
A square prism has dimensions of 12 cm by 12 cm by 15 cm as shown:
Calculate the length of HF, correct to two decimal places.
Calculate the length of DF, correct to two decimal places.
A rectangular prism has dimensions as labelled on the diagram.
Find the following lengths in surd form:
Find the area of \triangle ADG, in surd form.
Consider the triangular prism below:
Find, to two decimal places:
The value of x.
The value of y.
The value of z.
Consider the following rectangular prism:
Find the value of x to two decimal places.
Find the value of y to two decimal places.
A new diagonal has been added as shown, with length z cm:
Find the value of z to two decimal places.
Hence, find the area of the triangle bounded by the lengths a, y \text{ and } z.
A wooden plank of the greatest possible length is placed in a garden shed as shown in the diagram:
Calculate the length of the plank of wood correct to one decimal place.
This triangular prism shaped box labelled ABCDEF needs a diagonal support inserted as shown:
Write an expression for the length of BF in terms of BD and DF.
Hence, find the length of AF in terms of AB, BD and DF.
If AB = 19, BD = 30 and DF = 43, find the length of AF to two decimal places.
Find the length of AF be if AB, BD and DF increased by 10. Give your answer to two decimal places.
A triangular prism has dimensions as shown in the diagram:
Find the exact length of CE.
Find the exact length of BE.
Find the exact length of CX.
The following solid is a right pyramid with a square base. The pyramid has its apex, V, aligned directly above the centre of its base, W.
Calculate the length of AW, correct to two decimal places.
Hence, find the length of VW, correct to two decimal places.
Consider the cone with slant height of 13 m and perpendicular height of 12 m:
Find the length of the radius, r.
Hence, find the length of the diameter of the cone's base.
A right-angled triangular divider has been placed inside a box, as shown in the diagram:
Calculate the area of the triangular divider, correct to two decimal places.
A soft drink can has a height of 11 cm and a radius of 4 cm. Find L, the length of the longest straw that can fit into the can.
Round your answer down to the nearest cm, to ensure it fits inside the can.
A juice container has the shape of a rectangular prism. It needs a straw that must extend 20 mm beyond the container while touching the furthest corner of the base.
Find the exact length of the diagonal of the base, x.
Hence, find the length of the long diagonal of the juice container, z. Round your answer to two decimal places.
Hence, calculate the length of the straw needed. Round your answer to the nearest millimetre.
A builder needs to carry lengths of timber along a corridor in order to get them to where he is working. There is a right-angled bend in the corridor along the way. The corridor is 2 \text{ m} wide and the ceiling is 2.5 \text{ m} above the floor:
Calculate the maximum length of timber, in surd form, that would fit in the corridor when held parallel to the ground.
Since the corridor has a height of 2.5 \text{ m}, he can fit a longer piece of timber around the corner by angling it so that it reaches from the floor to the ceiling.
Find the maximum length of timber that is able to fit through the corridor, giving your answer correct to two decimal places.
