Pythagoras’ theorem in real life
Right-angled triangles are seen in many real life situations. Here are some problems you may come across and need to use Pythagoras to calculate lengths and distances.
Question
In a water sports competition, players are to sail 9 km south from the start of the race to buoy A and then sail 6 km east to buoy B before sailing back to the start. What is the total distance of the race?
1. Add all the details to the diagram:
2. We are calculating the hypotenuse so we need to add:
\({a}{^2}\) = 92 + 62
\({a}{^2}\) = 81 + 36
\({a}{^2}\) = 117
\({a}\) = \(\sqrt{117}\)
\({a}\) = 10.81665383
\({a}\) = 10.82 km (to two decimal places)
3. To find the total distance, add together the three lengths
Distance = 9 + 6 + 10.82
Distance = 25.82 km (to two decimal places)
Question
Brian wants to rest a 4 m long ladder against his garden shed so that it rests at the top of the wall which is 3 m high.
How far away from the base of the shed does he need to place the ladder?
1. Draw in the right angled triangle and add the lengths given:
2. We know the hypotenuse so we will need to subtract:
\({f}\)2 = 42 - 32
\({f}\)2 = 16 – 9
\({f}\)2 = 7
\({f}\) = \(\sqrt{7}\)
\({f}\) = 2.645751311
3. Round to two decimal places.
The bottom of the ladder needs to be 2.65 m from the base of the shed.
Question
Angharad is buying a new 32 inch screen TV. (Remember that TV screens are measured along the diagonal).
1. If it is 18 inches tall, how wide is the screen?
2. Her TV cabinet is 28 inches wide. Will the TV fit inside it?
1. We know the hypotenuse so we will need to subtract:
width2 = 322 - 182
width2 = 700
width = \(\sqrt{700}\)
width = 26.45751311 inches
2. The TV cabinet is 28 inches wide so yes, the TV will fit inside.
Question
The diagram shows a square set inside a circle.
The radius of the circle is 10 cm. Find the area of the square.
1. Identify the right-angled triangle:
2. We are finding the hypotenuse, so we need to add:
\({L}\)2 = 102 + 102
\({L}\)2 = 100 + 100
\({L}\)2 = 200
\({L}\) = \(\sqrt{200}\)
\({L}\) = 14.14213562
(Don’t round yet as we will need this value for the next calculation)
3. To find the area of the square we need to calculate:
\({L}~\times~{L}\)
14.14213562 × 14.14213562 = 200 cm2
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