naming sides and vertices of the triangle
just a small selection
RIGHT ANGLE TRIGONOMETRY
Pythagoras' theorem, also known as the Pythagorean theorem is used to calculate the lengths of the sides of right angled triangles. A right angled triangle has one angle which is 90 degrees, see fig. 1. or fig. 2, identified by the small square in the lower left hand corner. If we know the length of any of the two sides of a triangle, we can calculate the third unknown side of a triangle.
Pythagoras was a famous Greek mathematician and philosopher. Although the theorem is named after him, he probably didn't invent it as there is evidence to suggest many cultures used the theorem hundreds of years before Pythagoras. But there is a difference between using a method and proving it and so he is credited with proving it. The longer side of the triangle is called the hypotenuse, that's a Greek word just like Pythagoras.
SO WHAT DOES THE FORMULA MEAN?
The formula states that "The square of the hypotenuse (the longer side)" is equal to the sum of the squares of the two the shorter sides, (a and b).
A right angled triangle (called a triangle from now on) has a base length of 4cm and a vertical height of 3cm, calculate the length of the hypotenuse. Draw a diagram.
Set up the equation. Substitute the numbers into the formula. Square the numbers and add the results together. Don't forget to take the square root of c at the end. This particular triangle is called the 3,4,5, triangle for obvious reasons and is used in many practical applications. The maths used here is quite simple and we will look at how to do this with a calculator soon and show you a shortcut you can use if you wish.
This might seem a bit picky but the square root of 25 has 2 solutions, one positive as in the above and one negative as in -5cm. This is because -5 x -5 also equals 25. Technically the answer is +/- 5 cm and then you would state that +5 is the correct answer. This is something to keep in mind for later years.
Visually, this can be shown below in Fig.4.
How to use a calculator on the above equation
This method on the calculator is a short cut method for finding the length of a long side of a triangle. Put in the square root symbol, enclose the squares of the lengths of the sides in brackets. This enables the solution to be obtained quickly. Don't just write down the answer, show your working and don't forget the units (cm, m, km, kg etc.) and be able to round to 2 decimal places.
EXAMPLE CALCULATIONS
Exercise 1: Calculate the lengths of the hypotenuse in these triangles. Give your answer to 2 decimal places.
ANSWERS:
1) 10.30cm
2) 3.22m
3) 15.26m
4) 102.00cm
1) 10.30cm
2) 3.22m
3) 15.26m
4) 102.00cm
fIND THE SHORTER SIDE OF A RIGHT ANGLED TRIANGLE
To Find the short side of a right angle triangle, we rearrange the Pythagoras formula to make a or b the subject. Then follow the same procedure as above for finding the hypotenuse length. You can also use the shortcut method on the calculator.
Exercise 2:Calculate the lengths of the unknown shorter sides using Pythagoras' Theorem.
Answers to 2dp
1)95.70m
2)182.93
3)13.86m
4)63.11m
1)95.70m
2)182.93
3)13.86m
4)63.11m
Exercise 3(a) Given the roof height and base dimensions, calculate the total length of the roof of the Parthenon.
3(b) Calculate the approximate height of the roof beam given that the trusses form equilateral triangles.
A shed is being constructed on a work site. The builder decides to brace one of the wall frames with a piece of 4" x 2" timber. This will hold the structure in place while the remaining timber is nailed in place. From the measurements given calculate the minimum length of diagonal bracing securing the frame. The answer will be approximate in this case as is usual with a lot of practical examples.
The timber used in this construction work came from Mitre 10 Mega Lincoln Rd in 5m lengths, so it is often necessary to carry out calculations and cut timber on the ground at the construction site. However, the staff at most hardware stores will usually cut timber for minor DIY jobs to your requirements if your mate Jonesy can't help you.