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The top of a ladder touches the wall at a height of 6 feet. Find the length of the ladder if the length is two feet more than its distance from the wall.

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2 Answers

Picture this as a right triangle. The "distance from the wall" (i.e. the ground) is the bottom of the triangle, and the ladder is the hypotenuse. You are given the height as 6 ft, so this is the third side of the triangle.

Let x equal the ground / base of the triangle. 

Because this is a right triangle, we can use the Pythagorean theorem (a2 + b2 = c2), where a and b are the two sides making the right angle, and c is the hypotenuse.

We are given in the problem that x is two feet more than the ladder / hypotenuse. So c = x+2. One side of triangle can be a, and the other b. Let's say that a is the height, which is given as 6, and b is the ground, which we are calling x:


a=6                       b=x                c=x+2

Plug this into the Pythagorean theorem:    62 + x2 = (x+2)2

First, simplify the right side by expanding it:
(x+2)2 = (x+2)(x+2) = x2 + 4x + 4
62 + x2 = 36 + x2 = x2 + 4x + 4

Subtract 4 from both sides: 32 + x2 = x2 + 4x

Subtract x2 from both sides: 32 = 4x

Divide both sides by 4: 8 = x

Remember that x was equal to the distance from the wall, and the length of the ladder is two feet more than x.

Therefore, the length of the ladder = x+2 = 8+2 = 10 feet.

One leg is 6 ft, the other leg is 8 ft. By Pythagorean triple {6, 8, 10}, the hypotunese must be 10 ft, which is the length of the ladder.

Comments

Pythagorean triples are great if you understand their application. For someone that is unfamiliar with the concept, it might be confusing

 

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