Question

construction is under way at an airport. this map shows where the construction is taking place. if road a and road b are parallel, what is the distance from p to q on road c? the map has road c with segments 800 feet and 1,200 feet, a slanted road with 650 feet and a segment marked?? between points q and p (with p on road a and q on road b, road a and road b parallel).

Explanation:

Step1: Identify Similar Triangles

Since Road A and Road B are parallel, the triangles formed are similar by the Basic Proportionality Theorem (Thales' theorem). Let the distance from P to Q be \( x \). The sides of the triangles are proportional. The ratio of the sides of the smaller triangle to the larger triangle is \( \frac{800}{800 + 1200}=\frac{800}{2000}=\frac{2}{5} \)? Wait, no, actually, the segments on Road C are 800 and 1200, so the total length from the top to the bottom on Road C is \( 800 + 1200 = 2000 \) feet? Wait, no, looking at the diagram, the segment from the top of Road C to the intersection with Road A is 800 feet, and from there to the intersection with Road B is 1200 feet. The segment from the top of the slant road (Road C? Wait, no, the slant road is the one with 650 feet and the unknown. Wait, maybe the two parallel roads (Road A and Road B) create similar triangles with Road C. So the ratio of the corresponding sides should be equal. Let's denote the distance from P to Q as \( x \), and the length from the top of the slant road to P is 650 feet? Wait, no, the diagram shows that the segment from the top (where Road C meets the slant road) to P is 650 feet? Wait, no, maybe the length from the top of the slant road to Q is \( 650 + x \), and the segments on Road C are 800 and 1200. So the ratio of the sides: \( \frac{800}{800 + 1200}=\frac{650}{650 + x} \)? Wait, no, that might be reversed. Wait, actually, since Road A and Road B are parallel, the triangles are similar, so the ratio of the distance between Road A and Road B on Road C to the total distance on Road C is equal to the ratio of the distance between P and Q on the slant road to the total length of the slant road? Wait, maybe I got the segments wrong. Let's re-express:

Let the distance from P to Q be \( x \). The two parallel lines (Road A and Road B) cut Road C into segments of 800 ft and 1200 ft, and the slant road (let's call it Road D) into segments of 650 ft (from the top to P) and \( x \) (from P to Q). Wait, no, the top segment on Road D is 650 ft (from the top intersection with Road C to P), and the segment from P to Q is \( x \). So the ratio of the segments on Road C is \( \frac{800}{800 + 1200}=\frac{800}{2000}=\frac{2}{5} \), and the ratio on Road D should be the same. Wait, no, actually, the triangle formed by Road A, Road C, and the slant road is similar to the triangle formed by Road B, Road C, and the slant road. So the ratio of the corresponding sides is \( \frac{800}{800 + 1200}=\frac{800}{2000}=\frac{2}{5} \), and the ratio of the segments on the slant road should be \( \frac{650}{650 + x}=\frac{2}{5} \)? Wait, that would give \( 5 \times 650 = 2 \times (650 + x) \), \( 3250 = 1300 + 2x \), \( 2x = 1950 \), \( x = 975 \)? That doesn't seem right. Wait, maybe the other way: the ratio of the smaller triangle (with Road A) to the larger triangle (with Road B) is \( \frac{800}{1200}=\frac{2}{3} \)? No, that's not. Wait, perhaps the correct ratio is \( \frac{800}{1200}=\frac{2}{3} \), so the length from P to Q is \( 650 \times \frac{3}{2} \)? No, that would be 975, but that's the same as before. Wait, maybe I mixed up the segments. Let's look again:

Road C has two segments: 800 ft (from top to Road A) and 1200 ft (from Road A to Road B). Road D (the slant road) has a segment of 650 ft (from top to P, on Road A) and \( x \) (from P to Q, on Road B). Since Road A and Road B are parallel, the triangles are similar, so the ratio of the sides is equal. So \( \frac{800}{1200}=\frac{650}{x} \)? Wait, no, that would be \( \frac{800}{1200}=\frac{2}{3} \), so \( \frac{2}{3}=\frac{650}{x} \), \( x = 650 \times \frac{3}{2}=975 \). Wait, but that would mean the length from P to Q is 975? But let's check the total length: 650 + 975 = 1625, and the ratio of 800 to 2000 (800 + 1200) is 800/2000 = 2/5, and 650/1625 = 650 ÷ 1625 = 0.4 = 2/5. Oh! Wait, 800 + 1200 = 2000, so the ratio of the top segment (800) to the total (2000) is 800/2000 = 2/5. Then the segment from the top of the slant road to P is 650, which should be 2/5 of the total length of the slant road. Let the total length of the slant road be \( L \), so \( \frac{2}{5}L = 650 \), so \( L = 650 \times \frac{5}{2}=1625 \). Then the length from P to Q is \( L - 650 = 1625 - 650 = 975 \). Yes, that makes sense. So the distance from P to Q is 975 feet.

Wait, let's do it with proportions. Let \( x \) be the distance from P to Q. The two triangles are similar, so the ratio of corresponding sides is equal. The side of the smaller triangle (with Road A) is 800 ft, and the side of the larger triangle (with Road B) is \( 800 + 1200 = 2000 \) ft. The side of the smaller triangle on the slant road is 650 ft, and the side of the larger triangle on the slant road is \( 650 + x \) ft. So:

\[ \frac{800}{2000}=\frac{650}{650 + x} \]

Cross-multiplying:

\[ 800(650 + x)=2000 \times 650 \]

Simplify:

\[ 800 \times 650 + 800x = 2000 \times 650 \]

Subtract \( 800 \times 650 \) from both sides:

\[ 800x = (2000 - 800) \times 650 \]

\[ 800x = 1200 \times 650 \]

Solve for \( x \):

\[ x=\frac{1200 \times 650}{800} \]

Simplify \( \frac{1200}{800}=\frac{3}{2} \), so:

\[ x = \frac{3}{2} \times 650 = 975 \]

Step2: Verify the Proportion

Check the ratio: \( \frac{800}{2000}=\frac{2}{5} \), and \( \frac{650}{650 + 975}=\frac{650}{1625}=\frac{2}{5} \) (since 650 ÷ 1625 = 0.4 = 2/5). So the proportion holds.

Answer:

The distance from P to Q is \(\boxed{975}\) feet.