Scale Drawing and a Parallel Path
A landscape architect’s scale drawing shows a triangular courtyard ABC with AB = 6 cm, AC = 8 cm, and BC = 10 cm. The drawing uses a scale of 1 cm : 150 cm. A path DE is drawn parallel to BC, with D on AB and AD = 2 cm. The architect claims that because AD is one-third of AB, the path length DE is one-third of BC and the area of triangle ADE is one-third of the courtyard’s area. A contractor uses the drawing to order paving stones for the small triangular bed ADE and needs both the actual path length and the actual area. The contractor also questions whether the 6-8-10 triangle is right, because that affects whether a triangular shade sail can be installed at the corner. No angle measures are given on the drawing, and the scale must be applied consistently to lengths and areas.
1. Verify the architect’s claim about DE and the area of triangle ADE. Compute ADE’s actual area in square meters and compare it with one-third of the courtyard area.
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Because DE is parallel to BC, triangle ADE is similar to triangle ABC. The linear scale factor from ABC to ADE is AD/AB = 2/6 = 1/3. Therefore DE = (1/3)(BC) = (1/3)(10) = 10/3 cm on the drawing. The actual path length is (10/3)(150) = 500 cm = 5 m. Area does not scale by the same factor as length: it scales by the square of the linear scale factor, so area(ADE)/area(ABC) = (1/3)² = 1/9. The drawing’s courtyard area is (1/2)(6)(8) = 24 cm² because the 6-8-10 triangle is right. The actual courtyard area is 24 × 150² cm² = 540,000 cm² = 54 m². Thus triangle ADE has area 54/9 = 6 m². One-third of the courtyard area would be 18 m², so the architect’s area claim overestimates by a factor of 3.
2. Is triangle ABC right? Justify your answer and state which vertex is the right angle. Explain how the parallel path affects that conclusion for triangle ADE.
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Check the converse of the Pythagorean theorem: 6² + 8² = 36 + 64 = 100 = 10². Since the square of the longest side equals the sum of the squares of the other two sides, triangle ABC is right. The longest side is BC, so the right angle is opposite BC, at vertex A. Because DE is parallel to BC, triangle ADE is similar to triangle ABC by AA similarity: angle A is shared, and the corresponding angles at D and B are equal. Similar triangles have equal corresponding angles, so triangle ADE is also right, with its right angle at A.
3. Paving stones are sold in boxes covering 2.5 m² each, with 10% waste. Determine the number of boxes needed and critique the result of using the architect’s one-third area estimate.
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The actual area of triangle ADE is 6 m². Including 10% waste gives 6 × 1.10 = 6.6 m². Each box covers 2.5 m², so 6.6/2.5 = 2.64 boxes. Since boxes cannot be fractional, the contractor must buy 3 boxes. If the architect’s one-third estimate were used, the area would be 54/3 = 18 m². With waste, that becomes 18 × 1.10 = 19.8 m², requiring 19.8/2.5 = 7.92, or 8 boxes. That is a serious overorder: the correct area ratio is 1/9, not 1/3, because area scales as the square of the linear scale factor.