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Copland CollegeYear 11/12 - General Maths |
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Copland CollegeYear 11/12 - General Maths |
Pythagoras Spreadsheet Activity
Pythagoras' Theorem: Pythagoras' theorem states that if you square the two shorter sides in a right-angled triangle and add them together, you get the same as when you square the longest side (the hypotenuse). Remember SQUARING means multiplying a number by itself.
Pythagoras' formula: In a right-angled triangle, if a and b are the two shorter sides, and c is the hypotenuse, then a² + b² = c²
Task #1
You are required to design a PowerPoint presentation on Pythagoras (minimum of six slides). You need to focus on historical aspects, Pythagoras' Theorem itself and any applications it may have. Let me know when you have completed this task. This activity is part of your class mark. It needs to be completed today. Use the links below to find pictures and information
Useful links:
An Interactive Proof of Pythagoras' Theorem
Pythagorean Theorem and its many proofs
For this activity you will download a spreadsheet by clicking here. The spreadsheet allows you to alter the length of the sides of a right angled triangle. In doing so you are able to examine Pythagoras' Theorem and answer the following questions. The spreadsheet looks in part like the following screen shot. By clicking on the slider bars you are able to vary the length of the shorter sides of the triangle and then observe the alteration that occurs to the hypotenuse. Save the spreadsheet into your home folder before you start any work with it
Spreadsheet tasks to complete:
1) If one shorter side of a right-angled triangle on the Y axis is 7, the other one on the X axis is 8, what length is the hypotenuse of this right-angled triangle?
2) If one shorter side of a right-angled triangle on the Y axis is 3, the hypotenuse of this right-angles triangle is 5, what length is the other shorter side on the X axis?
3) If one shorter side of a right-angled triangle on the X axis is 6, the hypotenuse of this right-angles triangle is 10, what length is the other shorter side on the Y axis?
4) Is it possible to have a right-angled triangle with one of the shorter sides 15 units in length and the hypotenuse equal to 17 units in length?
Test yourself on Pythagoras by clicking on the following link: Pythagoras Questions
Similar Triangles
Using shadows produced by a metre stick to measure heights.
At some given time of the day a group of students measured the shadow produced by a metre stick held vertically on flat ground. The shadow length was 65 cm. That is a shadow length of 65 cm (0.65 m) was produced by a stick 1 metre in length. To determine the height of any other vertical object, at the same time of the day, we merely need to multiply the shadow length by the ratio 100/65.
This is very interesting and suggests that when we are dealing with triangles and the lengths of the their sides it is the ratios of the lengths of the sides that are important. In fact the ratios of the lengths of the sides are more important than the actual lengths taken by themselves. We will investigate this further by using a spreadsheet.
Open a new Excel spreadsheet. Your task is to set up a spreadsheet that will ,
a) accurately determine the height of an object knowing length if its shadow and knowing of the length of a shadow produced by a metre stick.
b) accurately determine the length of a shadow expected knowing the height of the object and knowing length of the shadow produced by a metre stick.
The following graphic will help you. Note here that the metre stick and shadow are measured in cm.
All numbers displayed in bold are generated by a formula not a typed in number. These formulas can be seen in the following.
Remember when you use the $ symbol it is called absolute referencing. This forces Excel to take the value in a specific cell each time the formula is copied. The alternative to absolute referencing is relative referencing. Relative referencing tells Excel to take a value displayed in a cell that is, for example two cells up and one cell to the right. Thus the cell referred to as you copy down a column is always different.
Set up another spreadsheet that is similar to the one that you have just completed. Use exactly the same format and formulas however the value in cell B4 will be 140 rather that 65. This time the measurements of shadow length were taken in the afternoon. It was found that at 4.00 pm the 1 metre stick produced a shadow of 1.40 m. Remember that 1.40 m is 140 cm.
Look at the spreadsheets that you have constructed and answer the following questions. Answer these on the computer. When you have finished call me over so that you name can be marked off.
Q1) How would you define a ratio?
Q2) The spreadsheets measure the heights of the metre rule and the shadow produced in centimetres. Could you use metres instead? Demonstrate whether this is possible using the spreadsheet.
Q3) If the shadow data was such that a 1 metre stick produced a shadow of 65 cm what is the height of an object if it has a shadow length of 20.5 metres? Remember the object is higher than the shadow. The question you need to ask yourself is do you multiply by 0.65/1.00 or 1.00/0.65. Which gives the larger number? Us the spreadsheet to help you.
Q4) If an object was 75 metres in height what would be the length of its shadow at 4.00 pm in the afternoon.
Getting a Tan:
For this activity you will need 3 metre sticks and a protractor. You should work in a group of two or three people. Set up your rulers flat on the floor. See diagram below, Use a protractor to measure the angle.
For each triangle measure the length of the opposite and adjacent sides for the angles, 20, 30, 45, 60, and 80 degrees. Use the protractor to measure the angles. The hypotenuse will always be 1 metre. Make the measurements as accurate as possible.
Set up a spreadsheet that will include the following data,
Remember that all calculations must be done by writing Excel formulas. For example the formula for the cell E4 (adjacent divided by hypotenuse) is =C4/B4 Remember Excel needs an equal sign to start a formula.
Using your calculator find values for sin, cos and tan for each of the angles shown above. To do this press the appropriate button, sin, cos or tan on your calculator and then the angle. Your calculator MUST be set to Degrees mode
Answer the following question.
1) What do you notice about the ratio opposite/hypotenuse and the sin of the various angles?
2) What do you notice about the ratio adjacent/hypotenuse and the cos of the various angles?
3) What do you notice about the ratio opposite/adjacent and the tan of the various angles?
4) The sin, tan and cos of a particular a particular angle is really just ..............................?