Book C: Pythagoras worksheet 2 - Solutions and Commentary
Introduction
Pythagoras' Theorem applies to right angled triangles. A right angled triangle has two sides \( a \) and \( b \), and a hypotenuse \( c \) that is always opposite the right angle. With that defined, we can use some formulas.
The picture below demonstrates the formula. The area of the attached square 'c' is the sum of the areas of the squares 'a' and 'b'.
The general form of the Pythagoras Theorem is the following:
\( c^2 = a^2 + b^2 \)
where \( c \) is the length of the hypotenuse, and \( a \) and \( b \) are the lengths of the other two sides.
NOTE: There is no particular "order" for \( a \) and \( b \). They are just the two sides that are not the hypotenuse.
The formula above is an implicit formula, meaning that \( c \) is defined implicitly as \( c^2 \).
Solving for the hypotenuse (\( c \))
The formula for \( c \) is as follows.
\( c = \sqrt{a^2 + b^2} \)
Normally, the square root precedes the \( a^2 + b^2 \). However, it is also possible to do the same thing using powers. Taking the square root if a number is the same as raising the number to the power of 0.5.
\( c = \left(a^2 + b^2\right)^{0.5} \)
This form may be easier to do on your calculator.
Solving for \( a \) and \( b \)
The formulas for \( a \) and \( b \) are obtained by rearranging the formula for \( c^2 \).
\( c^2 = a^2 + b^2 \rightarrow a^2 = c^2 - b^2 \)
The formula is similar for \( b^2 \): \( b^2 = c^2 - a^2 \)
The explicit formulas for \( a \) and \( b \) are as follows:
- \( a = \sqrt{c^2 - b^2} \)
- \( b = \sqrt{c^2 - a^2} \)
Question 10
In this case the sides appear to be given dimensions of millimetres, this is ok as long as all sides have the same dimensions when put into the formulas.
The side x is adjacent to the right angle, so it is not the hypotenuse.
The other side lengths \( b \) and \( c \) are 2 mm and 3 mm respectively.
The side \( a \) is solved using the explicit formula above.
\( a = \sqrt{c^2 - b^2} = \sqrt{3^2 - 2^2} = \sqrt{5} \) = 2.24 mm (2 dp) #.
Since the side lengths are given in mm, the answer will be in mm.
Question 11
In this case the sides appear to be given dimensions of metres, this is ok as long as all sides have the same dimensions when put into the formulas.
The side v is adjacent to the right angle, so it is not the hypotenuse.
The other side lengths \( b \) and \( c \) are 4 m and 6 m respectively.
The side \( a \) is solved using the explicit formula above.
\( a = \sqrt{c^2 - b^2} = \sqrt{6^2 - 4^2} = \sqrt{20} \) = 4.47 m (2 dp) #.
Since the side lengths are given in m, the answer will be in m.
Question 12
The side z is adjacent to the right angle, so it is not the hypotenuse.
The other side lengths \( b \) and \( c \) are 5 and 8 respectively.
The side \( a \) is solved using the explicit formula above.
\( a = \sqrt{c^2 - b^2} = \sqrt{8^2 - 5^2} = \sqrt{39} \) = 6.24 (2 dp) #.
Question 13
The side Y is adjacent to the right angle, so it is not the hypotenuse.
The other side lengths \( b \) and \( c \) are 9 and 12 respectively.
The side \( a \) is solved using the explicit formula above.
\( a = \sqrt{c^2 - b^2} = \sqrt{12^2 - 9^2} = \sqrt{63} \) = 7.94 (2 dp) #.
Question 14
The side v is adjacent to the right angle, so it is not the hypotenuse.
The other side lengths \( b \) and \( c \) are 4 m and 6 m respectively.
The side \( a \) is solved using the explicit formula above.
\( a = \sqrt{c^2 - b^2} = \sqrt{13^2 - 8^2} = \sqrt{105} \) = 10.25 m (2 dp) #.
Since the side lengths are given in m, the answer will be in m.
Question 15
In this case the sides appear to be given dimensions of millimetres, this is ok as long as all sides have the same dimensions when put into the formulas.
The side unknown side (the symbol is missing, it will be assigned \( a \)) is adjacent to the right angle, so it is not the hypotenuse.
The other side lengths \( b \) and \( c \) are 24 mm and 30 mm respectively.
The side \( a \) is solved using the explicit formula above.
\( a = \sqrt{c^2 - b^2} = \sqrt{30^2 - 24^2} = \sqrt{324} \) = 18 mm #.
Since the side lengths are given in mm, the answer will be in mm.
This type of right-angled triangle, with all integer sides, is known as a Pythagorean Triple.
Question 16
The side unknown side (the symbol is missing, it will be assigned \( a \)) is adjacent to the right angle, so it is not the hypotenuse.
The other side lengths \( b \) and \( c \) are 5.2 and 19 respectively.
The side \( a \) is solved using the explicit formula above.
\( a = \sqrt{c^2 - b^2} = \sqrt{19^2 - 5.2^2} = \sqrt{333.96} \) = 18.27 #.
Question 17
The side unknown side (the symbol is missing, it will be assigned \( a \)) is adjacent to the right angle, so it is not the hypotenuse.
The other side lengths \( b \) and \( c \) are 17 and 19.2 respectively.
The side \( a \) is solved using the explicit formula above.
\( a = \sqrt{c^2 - b^2} = \sqrt{19.2^2 - 17^2} = \sqrt{79.24} \) = 8.92 #.
Question 18
In this case the sides appear to be given dimensions of millimetres, this is ok as long as all sides have the same dimensions when put into the formulas.
The side unknown side (the symbol is missing, it will be assigned \( a \)) is adjacent to the right angle, so it is not the hypotenuse.
The other side lengths \( b \) and \( c \) are 1.7 mm and 2 mm respectively.
The side \( a \) is solved using the explicit formula above.
\( a = \sqrt{c^2 - b^2} = \sqrt{2^2 - 1.7^2} = \sqrt{1.11} \) = 1.05 mm #.
Since the side lengths are given in mm, the answer will be in mm.