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Difficulty: MediumPythagorean Theorem and Special Right Triangles

A rectangular park ABCDABCD has a length of 2020 meters and a width of 1515 meters. A straight walking path is built from corner AA to a point PP on the diagonal path BDBD such that the path APAP is perpendicular to BDBD. What is the length, in meters, of the path APAP?

Answer: 12 meters

Answer

The length of the path APAP is 1212 meters.
The diagonal BDBD divides the rectangular park into two congruent right triangles. For right triangle ABDABD, the legs are AB=20AB = 20 and AD=15AD = 15. Using the Pythagorean theorem, the hypotenuse BD=202+152=25BD = \sqrt{20^2 + 15^2} = 25 meters. The area of triangle ABDABD is 12×20×15=150\frac{1}{2} \times 20 \times 15 = 150 square meters. Since APAP is perpendicular to BDBD, APAP is the altitude to base BDBD, so the area can also be written as 12×25×AP\frac{1}{2} \times 25 \times AP. Equating the two areas, 12.5×AP=15012.5 \times AP = 150, which simplifies to AP=12AP = 12 meters.

Step-by-Step Solution

1
Calculate the length of the diagonal BDBD using the Pythagorean theorem on right triangle ABDABD.
BD=202+152=25BD = \sqrt{20^2 + 15^2} = 25 meters.
The diagonal forms the hypotenuse of the right triangle ABDABD, which is needed to calculate the altitude APAP.
2
Express the area of right triangle ABDABD using the two perpendicular legs, ABAB and ADAD.
Area=12×20×15=150\text{Area} = \frac{1}{2} \times 20 \times 15 = 150 square meters.
This establishes the total area of the triangle.
3
Express the area of the same triangle using the diagonal BDBD as the base and the perpendicular path APAP as the height.
Area=12×25×AP\text{Area} = \frac{1}{2} \times 25 \times AP.
This sets up an equation using the unknown path length APAP.
4
Equate the two area expressions and solve for APAP.
12.5×AP=150  ⟹  AP=1212.5 \times AP = 150 \implies AP = 12 meters.
Since both expressions represent the area of the same triangle, they must be equal.

Key Concept

Using the Pythagorean theorem to find the hypotenuse of a right triangle, and then using the area formula to find the altitude to the hypotenuse.

Alternative Method

Alternatively, you can use similar right triangles. Since triangle ABPABP is similar to triangle DBADBA, the ratio of their corresponding sides is equal: APAD=ABBD\frac{AP}{AD} = \frac{AB}{BD}. Substituting the known values gives AP15=2025\frac{AP}{15} = \frac{20}{25}, which simplifies to AP=15×45=12AP = 15 \times \frac{4}{5} = 12 meters.
Estimated Time:1m 30s
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Topics

Pythagorean Theorem and Special Right Triangles

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