Finding a Missing Leg with the Pythagorean Theorem

0:01A 13 metre ladder leans against a wall, and its top reaches 12 metres up.

0:07How far is the foot of the ladder from the wall?

0:11The upright wall meets the flat ground at a right angle, which is 90 degrees.

0:17So the wall, the ground and the ladder make a right triangle: a triangle with a right angle.

0:24The two sides that form the right angle are the legs. The wall is leg a, and the ground is leg b.

0:31The longest side, opposite the right angle, is the hypotenuse, c. Here it is the ladder.

0:39In a right triangle, the Pythagorean theorem says a squared plus b squared equals c squared. Squared means times itself.

0:48Put in what we know: 12 squared plus b squared equals 13 squared.

0:55We know the hypotenuse, so subtract 12 squared from both sides: b squared equals 13 squared minus 12 squared.

1:0413 times 13 is 169, and 12 times 12 is 144.

1:12169 minus 144 is 25, so b squared is 25.

1:20The square root undoes squaring. The square root of 25 is 5, because 5 times 5 is 25.

1:29So the foot of the ladder is 5 metres from the wall.

1:34Check: 5 squared plus 12 squared is 25 plus 144, which is 169, the same as 13 squared.

1:45A common mistake is adding instead: 169 plus 144 is 313.

1:54That is about 17.7 metres, longer than the 13 metre ladder, but a leg is always shorter than the hypotenuse.

2:03When the side you know is the hypotenuse, subtract the squares to find the missing leg.