Question

Which measurements could not represent the side lengths of a right triangle?A) 3cm, 4cm, 5cmB)3cm, 5cm, 9cmC)12cm, 16cm, 20cmD)16cm, 63cm, 65cm

Answer 1

The Pythagorean theorem states that for a rigth triangle, the square of the hypothenuse is equal to the sum of squares of the other two sides, symbolically:

`a^2+b^2=c^2`

To check if these sides lengths are of a rigth triangle you have to square them.

Remember that the hypothenuse is always the longest side.

So for the first set:

A)

3cm, 4cm and 5 cm

Lets take the side length 5cm as the hypothenuse

So a=3, b=4 and c=5

If the theorem checks then

`3^2+4^2=5^2`

Square all sides:

`\begin{gathered} 3^2=9 \\ 4^2=16 \\ 5^2=25 \end{gathered}`

Add both squared sides:

`9+16=25`

The result is equal to the square of the hypotenuse, this means that this side lengths corresponds to a rigth triangle.

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B)

a=3 cm

b=5 cm

c=9 cm (hypothenuse)

Square the three sides:

`\begin{gathered} a^2=3^2=9 \\ b^2=5^2=25 \\ c^2=9^2=81 \end{gathered}`

If the theorem checks then 9 + 25 must be equal to 81

`9+25=34`

The square sum of both sides is different from the quare of the hypotenuse, these side lengths do not correspond to a rigth triangle.

C)

a=12cm

b= 16 cm

c= 20 cm (hypothenuse)

Square the sides:

`\begin{gathered} a^2=12^2=144 \\ b^2=16^2=256 \\ c^2=20^2=400 \end{gathered}`

If the theorem checks then 144 plus 256 must be 400

`144+256=400`

The sum of squares of the sides is equal to the square of the hypothenuse, this set of side lengths belong to a right triangle.

D)

a=16cm

b=63cm

c=65cm (hypothenuse)

Square the sides:

`\begin{gathered} a^2=16^2=256 \\ b^2=63^2=3969 \\ c^2=65^2=4225 \end{gathered}`

If the theorem checks out, then 256 + 3969 must be equal to 4225:

`256+3969=4225`

The sum of squares of the sides is equal to the square of the hypothenuse, this set of side lengths belong to a right triangle.