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Triangular-based Hexagonal Circular-based pyramid,
pyramid,usually pyramid usually called a cone
called a tetrahedron
References: Altitude, Cone, Cube, Edge, Equilateral Triangle, Isosceles Triangle, Polyhedron, Regular Polyhedron, Right Pyramid, Vertex.
This theorem is named after the Greek mathematician Pythagoras, who lived about 500 years BC. The theorem applies to any right-angled triangle, and is a formula connecting the areas of squares that can be drawn on the sides of the triangle.
In figure a, the shaded triangle is right-angled, and squares A, 15, and C are drawn on the sides of the triangle. The theorem of Pythagoras states that
The theorem is usually written in the following form:
If the lengths of the sides of the triangle are a, b, and c units (see figure b), then
In words we say: ‚ÄėThe square on the hypotenuse of a right-angled triangle is equal to the sum of the squares of the other two sides.‚ÄĚ The theorem of Pythagoras is used for finding the length of one side of a right-angled triangle if the other two sides are known.
Figure c will enable you to demonstrate in a practical way the truth of the theorem of Pythagoras. You will need a copy of the figure and a pair of scissors. The square B has been divided up by drawing two lines each passing through the center of the square, with one of them parallel to the hypotenuse of the triangle. The two lines are perpendicular. Using scissors, remove the three squares from the triangle. Cut up
Area of square C = area of square A + area of square B
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square B into the four congruent pieces. Place the four pieces from B along with the square A onto the square C. These five pieces should jigsaw together to exactly cover the square C.
This demonstrates that the area of C = area of A + area of B.
A proof of the theorem of Pythagoras is given in the entry Triangle. The three examples that follow illustrate how to use this theorem to solve problems.
Example 1. A ladder rests on horizontal ground and leans against a vertical wall (see figure d). The foot of the ladder is 1.3 meters from the wall and reaches 2.9 meters up the wall. Find the length of the ladder to an accuracy of three decimal places.
Solution. We make a simple sketch of the right-angled triangle and its dimensions, calling the length of the ladder x meters. Write
x2 = 2.92 + 1.32 Theorem of Pythagoras: c2 = a2 + b2
x2 = 8.41 + 1.69 Using squares on a calculator
x2 = 10.1
x = Viol
x = 3.178 (to 3 dp) Using square roots on a calculator
The length of the ladder is 3.178 meters
Example 2. A ‚Äúflying fox‚ÄĚ consists of a wire stretched between two upright poles (see figure e). The poles are 10 meters apart. If the height of the shorter pole is
PYTHAGORAS‚Äô THEOREM 355
3 meters and the length of the wire is 12 meters, find the height of the longer pole. Assume that the wire has no ‚Äúsag‚ÄĚ and lies straight.
Solution. In order to use Pythagoras‚Äô theorem we need a right-angled triangle, and drawing a horizontal line from the top of the shorter pole to meet the longer pole forms this. Let the unknown length in the triangle be x meters. We can now apply Pythagoras‚Äô theorem to this triangle. Write
122 = x2 + 102 Pythagoras‚Äô theorem
144 = x2 + 100
144 - 100 = x2 Subtracting 100 from both sides of the equation
x2 = 44
x = -V44
x = 6.633 (to 3 dp) Using square roots on a calculator
The distance x must be added to the height of the 3-meter pole to obtain the height of the longer pole. The height of the longer pole is 9.633 meters.
Example 3. Amy works in a carpet warehouse. She stacks one roll of tufted carpet on top of two other rolls so that the rolls of carpet touch each other, as shown in figure f. The diameter of each roll is 1.4 meters. How high is the top of the tufted roll above the floor of the room?
Solution. We use Pythagoras‚Äô theorem in the right-angled triangle shown in figure f. The hypotenuse is of length 1.4 meters, which is equivalent to the diameter of the end of the carpet. The base of the triangle is of length 0.7 meters, which is
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the radius of the end of the carpet. Let the other side of the triangle be x meters. Write
1.96 = jc2 + 0.49 0.49 = jc2
jc2 = 1.47 jc = -\/L47 x = 1.212 (to 3 dp)
Subtracting 1.96 from both sides of the equation
Using a calculator for the square root
Height of the tufted carpet above the floor = 0.7 + x + 0.7
= 0.7+ 1.212 + 0.7 = 2.612 meters
The converse of the theorem of Pythagoras is as follows:
If the square on the longest side of a triangle is equal to the sum of the squares on the other two sides, then the triangle is right-angled.
Example. Darren is a groundsman and is laying out a new soccer pitch. He needs to make sure that the comers of the pitch are right angles. He gets three tape measures and pulls them out to lengths of 30, 40, and 50 meters. Using two other helpers, he tightly stretches the tapes to form a triangle. This triangle is right-angled, because 502 = 302 + 402.