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

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+256	Evaluate the squares
\displaystyle c^{2}	\displaystyle =	\displaystyle 400	Evaluate the sum
\displaystyle c	\displaystyle =	\displaystyle \sqrt{400}	Square root both sides
\displaystyle c	\displaystyle =	\displaystyle 20	Evaluate the square root

Example 2

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 100	Evaluate the squares
\displaystyle x^{2}+36-36	\displaystyle =	\displaystyle 100-36	Subtract 36 from both sides
\displaystyle x^{2}	\displaystyle =	\displaystyle 64	Evaluate the difference
\displaystyle x	\displaystyle =	\displaystyle \sqrt{64}	Square root both sides
\displaystyle x	\displaystyle =	\displaystyle 8	Evaluate the square root

Example 3

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 389	Evaluate the right side
\displaystyle l	\displaystyle =	\displaystyle \sqrt{389}	Square root both sides
\displaystyle l	\displaystyle \approx	\displaystyle 19.72	Evaluate 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}, \quad 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

AC9M9M03

solve spatial problems, applying angle properties, scale, similarity, Pythagoras’ theorem and trigonometry in right-angled triangles

AC9M9SP01

recognise the constancy of the sine, cosine and tangent ratios for a given angle in right-angled triangles using properties of similarity

6.01 Finding unknown side lengths with Pythagoras' theorem

Ideas

Pythagoras' theorem

Examples

Example 1

Example 2

Example 3

Outcomes

AC9M9M03

AC9M9SP01

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