Finding Rectangle Dimensions When Width Is Defined in Terms of Length
Apply the geometry problem-solving strategy when one dimension of a rectangle is described relative to the other, requiring both unknowns to be expressed through a single variable before substituting into the perimeter formula.
Problem: The width of a rectangle is two feet less than the length. The perimeter is feet. Find the length and width.
- Read: A rectangle has a perimeter of ft, and its width is feet less than its length.
- Identify: The length and width of the rectangle.
- Name: Since the width is described in terms of the length, let = the length. The phrase "two feet less than the length" translates to = the width. Draw and label the figure with length and width .
- Translate: Write the perimeter formula and substitute:
- Solve: Distribute: . Combine like terms: . Add to both sides: . Divide by : , so . The length is feet. Find the width: . The width is feet.
- Check: , and
- Answer: The length is feet and the width is feet.
When neither dimension is given directly but one is described relative to the other, both unknowns must be expressed through a single variable before substituting into the formula. This produces an equation with like terms () that must be combined before isolating the variable — a key step that distinguishes this type of problem from ones where one dimension is already a known number. Note that the figure is not drawn until Step 3, after the variable expression for the width is established, so that one side can be labeled with the expression .
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