6. 2D Geometry

Move the sliders to adjust the leg lengths of a right triangle.

What do you notice about the sum of the area of the squares of the two shorter sides and the area of the square of the hypotenuse?

If you subtracted the areas of the shorter sides' squares from the area of the largest side's square, what do you get?

Can you find another set of sides, where all three sides have integer values?

Try to write an equation that will always be true using the variables a,\,b,\, and c.

The **Pythagorean theorem** says that in a right triangle the square of the **hypotenuse** is equal to the sum of the squares of the other two sides:

\displaystyle a^2+b^2=c^2

\bm{c}

is the length of the hypotenuse

\bm{a}

is one of the shorter side lengths

\bm{b}

is the other shorter side length

We can use this formula to find the length of any side of a right triangle if we know the lengths of the other two.

The **converse of the Pythagorean Theorem** says that if the square of the length of the hypotenuse equals the sum of the squares of the legs in a triangle, then the triangle is a right triangle. This can be used to determine whether a triangle is a right triangle given the measures of its three sides.

If a group of 3 integers can create the sides of a right triangle, we call this group of numbers a **Pythagorean triple**. This group of numbers (a,\,b,\,c) will satisfy the Pythagorean theorem: a^2+b^2=c^2.

(6,\,8,\,10) is a Pythagorean triple because:

If we know one Pythagorean triple we can create others using multiplication or division. 6, \,8 and 10 have a common factor of 2. If we divide each number in the triple by this common factor, we get another Pythagorean triple (3,\,4,\,5).

Multiplying (3,\,4,\,5) by 5 gives us another Pythagorean triple (15,\,20,\,25).

Determine if the triangle below is a right triangle. If so, label the hypotenuse and legs.

Worked Solution

Find the length of the hypotenuse, c in this triangle.

Worked Solution

Calculate the value of a in the triangle below.

Worked Solution

A triangle has side lengths of 6,\,7,\, and 11. Do these three side lengths make a Pythagorean triple?

Worked Solution

Luke knows the two largest numbers in a Pythagorean Triple, which are 37 and 35. What number, a, does Luke need to complete the triple?

Worked Solution

A sports association wants to redesign the trophy they award to the player of the season. The front view of one particular design is shown in the diagram:

a

Find the value of x.

Worked Solution

b

Find the value of y, rounded to two decimal places.

Worked Solution

Idea summary

The Pythagorean theorem states that in a right triangle the square of the **hypotenuse** (the side opposite the right angle) is equal to the sum of the squares of the other two sides:

\displaystyle a^2+b^2=c^2

\bm{c}

is the length of the hypotenuse

\bm{a}

is one of the shorter side lengths

\bm{b}

is the other shorter side length

A **Pythagorean triple** is an ordered triple (a,\,b,\,c) of three positive integers that represent the side lengths of a right triangle.