A narrow street is lined with tall buildings. The base of a 20 foot long ladder is resting at the base of the building on the right side of the street and leans on the building on the left side. The base of the 30 foot long ladder is resting at the base of the building on the left side of the street and leans on the building on the right side. The point where the two ladders cross is exactly 8 feet from the ground. Approximately, how wide is the street (to the nearest tenth)?

To solve this problem, you will need to find similar triangles within the problem. To begin, we will label the bases of our triangles created by the ladders, the ground, and a line perpendicular to the ground and the point where the two ladders meet. This will create two new triangles, and these new triangles are similar to the triangles created by the ladder and the ground. The blue triangle is similar to the triangle with the 20 ft. hypotenuse, and the green triangle is similar to the triangle with a 30 ft. hypotenuse. Assign a variable to both of the bases of these new triangles.

Using the Pythagorean Theorem, we can find the height of the triangles created by the ladders and the road. We can find them to be

for the smaller and larger triangle respectively.

Prove triangles are similar.
Notice that the blue triangle is similar to the triangle with a hypotenuse of 20, because they share one angle and have right angles.
Using our knowledge off similar triangles, we can create the proportions.

By isolating a and b in these formulas, and noticing in our image that a+b=x, we can find the formula

This will give us our solution to the problem, 16.212

Approximately, how wide is the street (to the nearest tenth)?

To solve this problem, you will need to find similar triangles within the problem. To begin, we will label the bases of our triangles created by the ladders, the ground, and a line perpendicular to the ground and the point where the two ladders meet. This will create two new triangles, and these new triangles are similar to the triangles created by the ladder and the ground. The blue triangle is similar to the triangle with the 20 ft. hypotenuse, and the green triangle is similar to the triangle with a 30 ft. hypotenuse. Assign a variable to both of the bases of these new triangles.

Using the Pythagorean Theorem, we can find the height of the triangles created by the ladders and the road. We can find them to be

for the smaller and larger triangle respectively.

Prove triangles are similar.

Notice that the blue triangle is similar to the triangle with a hypotenuse of 20, because they share one angle and have right angles.

Using our knowledge off similar triangles, we can create the proportions.

By isolating a and b in these formulas, and noticing in our image that a+b=x, we can find the formula

This will give us our solution to the problem, 16.212