Question

1. in the diagram, a person who is 6 ft tall is standing on the ground 3 ft away from point p. a line segment drawn from the top corner of the building to point p creates two similar triangles.

diagram of a building, a person, and point p with labels: person is 6 ft tall, 3 ft from p; building is 18 ft from p; height of building is h; not drawn to scale
which proportion can be used to find h, the height of the building in feet?
options:
\\(\frac{3}{h} = \frac{18}{6}\\)
\\(\frac{6}{3} = \frac{h}{18}\\)
\\(\frac{3}{18} = \frac{h}{6}\\) (corrected from original ocr for clarity, original had \\(\frac{3}{15} = \frac{h}{6}\\) which likely has a typo, but preserving as per ocr: \\(\frac{3}{15} = \frac{h}{6}\\))
\\(\frac{6}{3} = \frac{18}{h}\\)

Explanation:

Step1: Identify Similar Triangles

The two triangles (person's triangle and building's triangle) are similar, so corresponding sides are proportional. The person's height is 6 ft, base 3 ft. The building's height is \( h \), base 18 ft? Wait, no—wait, the distance from P to the building is 18 ft, and from P to the person is 3 ft. So the smaller triangle (person) has height 6, base 3. The larger triangle (building) has height \( h \), base 18? Wait, no—wait, the horizontal distance from P to the building is 18 ft, and from P to the person is 3 ft, so the base of the larger triangle is 18 ft, and the base of the smaller is 3 ft. The height of the smaller is 6, height of larger is \( h \). Wait, but wait, maybe the horizontal distance from the person to the building is \( 18 - 3 = 15 \) ft? Wait, no, the diagram: P to the person is 3 ft, person to the building's base is \( 18 - 3 = 15 \) ft? Wait, no, the total distance from P to the building is 18 ft. So the smaller triangle (person) has legs 6 (height) and 3 (base from P to person). The larger triangle (building) has legs \( h \) (height) and 18 (base from P to building). Wait, but similar triangles: corresponding sides. So the ratio of height to base for the smaller triangle should equal the ratio for the larger. So \( \frac{6}{3} = \frac{h}{18} \). Let's check the options. The options are:

1. \( \frac{3}{h} = \frac{18}{6} \)
2. \( \frac{6}{3} = \frac{h}{18} \)
3. \( \frac{3}{15} = \frac{h}{6} \) (no, 15 is 18-3, but that's not correct)
4. \( \frac{6}{3} = \frac{18}{h} \) (no)

Wait, wait, maybe I misread the base. Wait, the person is 3 ft from P, and the building is 18 ft from P. So the smaller triangle (person) has height 6, base 3. The larger triangle (building) has height \( h \), base 18. So similar triangles: \( \frac{\text{height of small}}{\text{base of small}} = \frac{\text{height of large}}{\text{base of large}} \), so \( \frac{6}{3} = \frac{h}{18} \). Let's check the options. The second option is \( \frac{6}{3} = \frac{h}{18} \), which matches. Wait, but let's check the other options. Wait, maybe the horizontal distance from the person to the building is 15 ft (18-3), so the larger triangle's base is 15 ft? Wait, no, the building's base is from P to the building, which is 18 ft. Wait, the diagram: P is a point, person is 3 ft from P, building is 18 ft from P. So the two triangles are similar, so corresponding sides. So height of small (6) over base of small (3) equals height of large (h) over base of large (18). So \( \frac{6}{3} = \frac{h}{18} \), which is the second option. Wait, but let's check the options again. The options are:

• \( \frac{3}{h} = \frac{18}{6} \)
• \( \frac{6}{3} = \frac{h}{18} \)
• \( \frac{3}{15} = \frac{h}{6} \) (15 is 18-3)
• \( \frac{6}{3} = \frac{18}{h} \)

Wait, maybe I made a mistake. Let's think again. The two triangles are similar, so the ratio of corresponding sides. The smaller triangle: height 6, base 3 (from P to person). The larger triangle: height \( h \), base 18 (from P to building). So \( \frac{6}{3} = \frac{h}{18} \), which is the second option. So that's the correct proportion.

Step2: Verify the Proportion

For similar triangles, the ratio of corresponding sides is equal. The height of the smaller triangle (person) is 6 ft, and its base (horizontal distance from P) is 3 ft. The height of the larger triangle (building) is \( h \), and its base (horizontal distance from P) is 18 ft. Therefore, the proportion is \( \frac{6}{3} = \frac{h}{18} \), which matches the second option (the one with \( \frac{6}{3} = \frac{h}{18} \)).

Answer:

The correct proportion is \( \boldsymbol{\frac{6}{3} = \frac{h}{18}} \) (the second option, e.g., if options are labeled, say B. \( \frac{6}{3} = \frac{h}{18} \)).