Question

how high up a vertical wall will a 26-foot-long extension ladder reach if its base is placed 10 feet away from the wall

Answer 1

24ft high

Explanation:

Assuming the wall is perpendicular to the ground (so that it forms a right angle, and that when the ladder is placed on the ground and leaned up against the wall that it forms a right triangle), we can use the Pythagorean Theorem to describe the situation mathematically.

The Pythagorean Theorem

For any right triangle (a triangle that has a right angle) the Pythagorean Theorem applies. The Pythagorean Theorem describes the relationship between side lengths as
`a^2+b^2=c^2`, where "c" is the length of the "hypotenuse" (the side across from the right angle), and "a" and "b" are the lengths of the "legs" (the other two sides that are touching the right angle). Since addition is commutative, it doesn't matter which leg you pick for side "a" or side "b", it only matters that the hypotenuse must be "c".

Given that the ladder is across from the right angle (the angle formed by the wall and the ground... the two "legs"), the ladder represents the hypotenuse in this situation.

Substituting known values, and solving the equation

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

Substitute known values, and simplify...

`a^2+(10)^2=(26)^2`

`a^2+100=676`

Subtract 100 from both sides...

`(a^2+100)-100=(676)-100`

`a^2=576`

Apply the square root property...

`√(a^2)=\pm √(576)`

`a=\pm 24`

`a=24` or
`a=- 24`

Since the distance height the ladder will reach is not a negative number, we reject the negative solution. Hence, the ladder will reach 24ft high, if placed 10ft from the base of the wall.