Calculating the Length of the Hypotenuse and Unknown Side Length

To find the length of the hypotenuse in a right triangle when given the two other sides, you can use the Pythagorean theorem, which states:

[ c = sqrt{a^2 + b^2} ]

where ( c ) is the length of the hypotenuse, and ( a ) and ( b ) are the lengths of the two other sides.

1. For the first triangle:

– ( a = 12 ) cm

– ( b = 5 ) cm

[ c = sqrt{12^2 + 5^2} = sqrt{144 + 25} = sqrt{169} = 13 ] cm

So, the length of the hypotenuse is 13 cm.

2. For the second triangle, to find the unknown side length:

– Given the hypotenuse ( c = 9 ) in and one side ( a = 7 ) in, we rearrange the Pythagorean theorem to find ( b ):

[ b = sqrt{c^2 – a^2} ]

Plugging in the values:

[ b = sqrt{9^2 – 7^2} = sqrt{81 – 49} = sqrt{32} approx 5.7 ] in

Therefore, the unknown side length is approximately 5.7 in when rounded to the nearest