A wall 10 feet high is 8 feet away from a house. Find the length of the shortest ladder that will reach the house when one end rests on the ground outside the wall.

So in here. I did the similar triangle property. so i got y=10(x+8)/x . And i substituted it in the PYTHAGOREAN Theorem.
z^2= (the equation y above)^2 + (8+x)^2. Please help, i don't know what to do from here.

http://www.mathalino.com/reviewer/differential-calculus/14-ladder-reaching-house-ground-outside-wall

To solve for the length of the shortest ladder, we can follow these steps:

Step 1: Apply the similar triangle property to establish a relationship between the unknown lengths. You correctly obtained the equation y = 10(x + 8) / x, which relates the height of the wall (y) to the distance from the house (x).

Step 2: Substitute the expression for y into the Pythagorean theorem to solve for the ladder length (z). The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.

So, we have:

z^2 = y^2 + (8 + x)^2

Now we can substitute the expression for y that we derived in Step 1:

z^2 = (10(x + 8) / x)^2 + (8 + x)^2

Step 3: Simplify the equation. We can start by multiplying out the terms:

z^2 = (100(x + 8)^2 / x^2) + (8 + x)^2

Next, we need to find a common denominator for the fractions:

z^2 = (100(x + 8)^2 + x^2(8 + x)^2) / x^2

Now, expand and simplify:

z^2 = (100(x^2 + 16x + 64) + (8 + x)^2(x^2 + 16x + 64)) / x^2

z^2 = (100x^2 + 1600x + 6400 + (8x + x^2)(x^2 + 16x + 64)) / x^2

z^2 = (100x^2 + 1600x + 6400 + 8x^3 + 128x^2 + 512x + x^4 + 16x^3 + 64x^2) / x^2

Step 4: Combine like terms and simplify further:

z^2 = (x^4 + 24x^3 + 292x^2 + 2112x + 6400) / x^2

Finally, we have our equation for the ladder length. From here, you can solve for z by taking the square root of both sides:

z = sqrt((x^4 + 24x^3 + 292x^2 + 2112x + 6400) / x^2)

This expression will give you the length of the shortest ladder needed to reach the house, given the distance from the wall (x).

To find the length of the shortest ladder that will reach the house, you are correct to use the Pythagorean theorem. Let's go through the steps to solve the equation and find the length of the ladder.

1. Start with the equation you derived from the similar triangle property: y = (10(x + 8)) / x.

2. Substitute this value of y into the Pythagorean theorem equation: z^2 = y^2 + (8 + x)^2.

3. Expand and simplify the equation:
z^2 = [(10(x + 8)) / x]^2 + (8 + x)^2.
z^2 = (100(x + 8)^2) / x^2 + (8 + x)^2.

4. Simplify further by getting rid of the fraction:
z^2 = (100(x^2 + 16x + 64)) / x^2 + (8 + x)^2.
z^2 = (100x^2 + 1600x + 6400) / x^2 + (8 + x)^2.

5. Combine the terms in the numerator to get a common denominator:
z^2 = (100x^2 + 1600x + 6400 + x^2) / x^2 + (8 + x)^2.
z^2 = (101x^2 + 1600x + 6400) / x^2 + (8 + x)^2.

6. Multiply out (8 + x)^2:
z^2 = (101x^2 + 1600x + 6400) / x^2 + (64 + 16x + x^2).
z^2 = (101x^2 + 1600x + 6400 + x^2) / x^2 + (64 + 16x + x^2).

7. Combine like terms in the numerator:
z^2 = (102x^2 + 1600x + 6400) / x^2 + (64 + 16x + x^2).

Now, you can simplify the equation further or solve for the length of the ladder using another method, such as factoring or taking the square root of both sides.