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5 Trigonometry
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Level 10

5.01 Pythagoras' theorem

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Lesson

Pythagoras' theorem

A right angled triangle with hypotenuse and right angle labelled and opposite each other.

In a right-angled triangle the largest angle in the triangle is 90\degree. The side across from the right angle will be the largest side. We call this side the hypotenuse.

A right angled triangle with sides a, b and c on the hypotenuse. The Pythagoras theorem is written next to it.

All three sides of a right-angled triangle are related by the equation shown.

The two smaller sides will be called a and b, and the hypotenuse (the longest side) will be c.

This relationship between sides in a right-angled triangle is called Pythagoras' theorem. We can use this theorem to find both hypotenuses and short sides.

The most important thing to remember when finding a short side is that the two lengths need to go into different parts of the formula.

If we get the lengths around the wrong way, we will probably end up with the square root of a negative number (and a calculator error).

Examples

Example 1

Use Pythagoras' theorem to determine whether this is a right-angled triangle.

A right angled triangle with side lengths of 9, 16, and 18.
a

Let a and b represent the two shorter side lengths. First find the value of a^{2}+b^{2}.

Worked Solution
Create a strategy

Substitute the lengths of the two shorter sides and add them.

Apply the idea
\displaystyle a^{2}+b^{2}\displaystyle =\displaystyle (9)^{2} + (16)^{2}Substitute the value of a and b.
\displaystyle =\displaystyle 81+256Evaluate
\displaystyle =\displaystyle 337Evaluate the sum
b

Let c represent the length of the longest side. Find the value of c^{2}.

Worked Solution
Create a strategy

Substitute the length of the longest side.

Apply the idea
\displaystyle c^{2}\displaystyle =\displaystyle (18)^{2}Substitute the value of c
\displaystyle =\displaystyle 324Evaluate
c

Is the triangle a right-angled triangle?

Worked Solution
Create a strategy

Compare the results from parts (a) and (b) to determine if a^2+b^2=c^2.

Apply the idea

337\neq 324

The triangle is not a right-angled triangle because a^2+b^2 \neq c^2.

Example 2

Find the length of the unknown side c in the triangle below.

A right angled triangle with 2 short side lengths of 16 and 12, and the longest side length of c.
Worked Solution
Create a strategy

We can use the Pythagoras' theorem: c^{2}= a^{2}+b^{2} .

Apply the idea
\displaystyle c^{2}\displaystyle =\displaystyle a^{2}+b^{2}Write the formula
\displaystyle c^{2}\displaystyle =\displaystyle 12^{2}+16^{2}Substitute a and b
\displaystyle c^{2}\displaystyle =\displaystyle 144+256Evaluate the squares
\displaystyle c^{2}\displaystyle =\displaystyle 400Evaluate the sum
\displaystyle c\displaystyle =\displaystyle \sqrt{400}Square root both sides
\displaystyle c\displaystyle =\displaystyle 20Evaluate the square root

Example 3

Find the length of the unknown side x in the triangle below.

A right-angled triangle with 2 short side lengths of 6 and x, and long side length of 10.
Worked Solution
Create a strategy

Use Pythagoras' theorem: a^{2}+b^{2}=c^{2}.

Apply the idea

We can substitute the following a=x,\, b=6, and c=10 into the formula:

\displaystyle a^{2}+b^{2}\displaystyle =\displaystyle c^{2}Write the formula
\displaystyle x^{2}+6^{2}\displaystyle =\displaystyle 10^{2}Substitute the values
\displaystyle x^{2}+36\displaystyle =\displaystyle 100Evaluate the squares
\displaystyle x^{2}+36-36\displaystyle =\displaystyle 100-36Subtract 36 from both sides
\displaystyle x^{2}\displaystyle =\displaystyle 64Evaluate the difference
\displaystyle x\displaystyle =\displaystyle \sqrt{64}Square root both sides
\displaystyle x\displaystyle =\displaystyle 8Evaluate the square root

Example 4

A movie director wants to shoot a scene where the hero of the film fires a grappling hook from the roof of one building to the roof of another.

If the first building is 37 m tall, the other building is 54 m tall and the street between them is 10 m wide, what is the minimum length l of rope needed for the grappling hook? Round your answer to two decimal places.

The image shows two buildings with heights of 54 meters and 37 meters. Ask your teacher for more information.
Worked Solution
Create a strategy

Use the Pythagoras' theorem: c^{2}= a^{2}+b^{2} .

Apply the idea

Notice that we have a right-angled triangle, and the length of the grappling hook's rope will be the hypotenuse. The one short side is the width of the street, and the other short side is the difference in heights between the two buildings.

We can substitute the following a=10,\, b=(54-37), and c=l into the formula:

\displaystyle c^{2}\displaystyle =\displaystyle a^{2}+b^{2}Write the formula
\displaystyle l^{2}\displaystyle =\displaystyle 10^{2}+(54-37)^{2}Substitute the values
\displaystyle l^{2}\displaystyle =\displaystyle 389Evaluate the right side
\displaystyle l\displaystyle =\displaystyle \sqrt{389}Square root both sides
\displaystyle l\displaystyle \approx\displaystyle 19.72Evaluate the square root
Idea summary
A right angled triangle with sides a and b and c on the hypotenuse. The Pythagoras theorem formula is written next to it.

Pythagoras' theorem relates the three sides of a right-angled triangle, a and b are the two smaller sides, and the longest side, called the hypotenuse, is c.

We can also test to see if a triangle is right-angled by checking to see if its three sides satisfy a^{2}+b^{2}=c^{2}.

To find the hypotenuse:c^{2}=a^{2}+b^{2}

To find a shorter side use a^{2}=c^{2}-b^{2} or b^{2}=c^{2}-a^{2}.

We can take the square root of both sides to give us the following formulas:

c=\sqrt{a^{2}+b^{2}}, \quad a=\sqrt{c^{2}-b^{2}}, \quad b=\sqrt{c^{2}-a^{2}}

Outcomes

VCMMG346

Solve right-angled triangle problems including those involving direction and angles of elevation and depression.

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