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Difficulty: MediumRight Triangles and the Pythagorean Theorem

A rectangular field has a diagonal path of length 4040 meters. The length of the field is 88 meters greater than its width. What is the width, in meters, of the field?

Answer: 24 meters

Answer

The correct answer is 24.
Letting ww represent the width of the rectangular field in meters, the length is w+8w + 8 meters. The diagonal forms the hypotenuse of a right triangle with the width and the length as legs. By the Pythagorean theorem, w2+(w+8)2=402w^2 + (w+8)^2 = 40^2. Expanding this yields 2w2+16w+64=16002w^2 + 16w + 64 = 1600, which simplifies to w2+8w−768=0w^2 + 8w - 768 = 0. Factoring this equation gives (w−24)(w+32)=0(w-24)(w+32) = 0. Since width must be positive, the width is 24.

Step-by-Step Solution

1
Represent the dimensions of the rectangular field.
Let the width of the field be ww meters, so the length is w+8w + 8 meters.
The problem states the length is 8 meters greater than the width.
2
Apply the Pythagorean theorem.
w2+(w+8)2=402w^2 + (w + 8)^2 = 40^2
The diagonal of a rectangle forms a right triangle with the width and length as its legs.
3
Simplify the quadratic equation.
w2+8w−768=0w^2 + 8w - 768 = 0
Expanding the equation gives 2w2+16w−1536=02w^2 + 16w - 1536 = 0, which simplifies when divided by 2.
4
Solve for the variable ww.
(w−24)(w+32)=0(w - 24)(w + 32) = 0, so w=24w = 24 or w=−32w = -32.
Factoring the quadratic equation gives the possible values for the width.
5
Determine the valid physical width.
w=24w = 24
Since the width of a field must be positive, we reject the negative solution.

Key Concept

Using the Pythagorean theorem to relate the sides of a right triangle in a geometric word problem.
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Topics

Right Triangles and the Pythagorean Theorem

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