A wire is attached to the top of a pole. The pole is 2 ft shorter than the wire, and the distanc

step 1 Well, the problem can be drawn like this. The length of the wire is X. The length of the pole is

A wire is attached to the top of a pole. The pole is 2 ft...

A wire is attached to the top of a pole. The pole is $2 \mathrm{ft}$ shorter than the wire, and the distance from the wire on the ground to the bottom of the pole is 9 ft less than the length of the wire. Find the length of the wire and the height of the pole.

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Key Concepts

Quadratic Equations

Equations in which the highest power of the variable is two. They commonly appear when applying the Pythagorean theorem in algebraic form, especially after substituting expressions from one equation into another, thereby requiring techniques for finding their roots.

System of Equations

When a problem provides multiple relationships among unknown quantities, these can be expressed as simultaneous equations. Solving such a system allows for the determination of the unknown values that satisfy all conditions in the problem.

Pythagorean Theorem

A fundamental relation in Euclidean geometry stating that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. This theorem is essential for solving problems involving right triangles, such as determining the distance relationships between the wire, the pole, and the ground.

Algebraic Translation of Word Problems

The process of defining variables for unknown quantities and converting the verbal description of relationships into algebraic equations. This skill is key to setting up the problem correctly so that geometric conditions can be expressed mathematically.

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