A 10-foot ladder is leaning up a wall. The distance the ladder is from the wall is 2 feet shorter than the distance the ladder reaches on the wall. What is th

Question

 Algebra 1 

Answer 1

       |\

       |  \

    x |    \  10

       |      \

       |       \

        -------

          x-2

 

You have a right triangle.  The hypotenuse is 10 ft

The height on wall is x, the other side is x-2

 

By Pythagorean Theorem:

x² + (x-2)² = 10²

 

x² + x² -4x + 4 = 100

2x² - 4x - 96 = 0

 

Simplify by dividing everything by 2

 

x² - 2x - 48 = 0

 

We can factor this.  The factors of -48 that add to -2 are (-8)(6)

Replace -2x with -8x + 6x

 

x² - 8x + 6x - 48 = 0

 

x(x-8) + 6(x-8) = 0

(x+6)(x-8) = 0

 

x = -6   or   x = 8

 

Discard the negative.  x=8 ft

 

The top of the ladder hits the wall 8 ft above the ground.

The distance from bottom of wall to base of ladder is 6 ft

 

8² + 6² = 10²

64 + 36 = 100

100 = 100

Answer checks.

 

Answer 2

If you'll allow me to speculate on the missing part of your question:

What is the ladder's distance from the wall (at ground level)?

 

you have described a right triangle having hypotenuse 10 ft, horizontal leg h-2, and vertical leg h.

 

Using the Pythagorean theorem: 10² = (h-2)² + h²

 

100 = h² -4h + 4 + h² after squaring the binomial properly.  Then

96 = 2h² - 4h  and the next step will be to divide each term by 2, giving

 

h² -2h - 48 = 0   (h-8)(h+6) = 0 after factoring.  Therefore h = 8 ft (the root -6 is inapplicable, cannot be a length).

 

Predictably, the right triangle is a 6-8-10 triangle (dilation of the 3-4-5 triangle).

Answer 3

I am not sure what the question is.  I'll assume that you want to know how far up the wall the ladder reaches.  If that is the case, then proceed as follows:

 

Let x = distance of ladder from the floor to the top of the ladder

 

Then, x-2 = distance from bottom of ladder to the wall

 

Assuming that the floor is horizontal and the wall is vertical, we have a right triangle with the ladder as the hypotenuse.  

 

By the Pythagorean Theorem, x² + (x-2)² = 10²

 

Simplifying, we get x² + x² - 4x + 4 = 100

 

                         2x² - 4x - 96 = 0

 

                          x² - 2x - 48 = 0

 

                         (x-8)(x+6) = 0

 

                         x = 8 or x = -6

 

Since x cannot be negative, x = 8 feet