Question

pierre stands on one side of a river and dahlia stands directly across the river from him. pierre is 30 m from the river and dahlia is 50 m from the river, as shown in the figure.
each person walks away from the other in opposite, parallel directions. pierre walks 360 m and dahlia walks 40 m. they turn so that they can see each other and the side of the river nearest dahlia in the same line of sight.
what is the distance across the river at the point where pierre and dahlia begin?
m

Explanation:

Step1: Identify Similar Triangles

The two triangles formed (one with Pierre's path and the river, one with Dahlia's path and the river) are similar by AA similarity (right angles and same angle from the line of sight). Let the river width be \( x \) meters.

Step2: Set Up Proportion

For similar triangles, the ratios of corresponding sides are equal. So, \(\frac{360}{40}=\frac{30 + x}{50}\) (wait, no, correct sides: Pierre's horizontal distance is 360, Dahlia's horizontal is 40; Pierre's vertical from river edge is 30, Dahlia's is 50, and river width is \( x \). Wait, correct proportion: \(\frac{360}{40}=\frac{30 + x}{50}\)? No, re - examine. The triangles: one triangle has base 360, height \( 30 + x \)? Wait, no, the river width is \( x \), Pierre is 30m from river, Dahlia is 50m from river. The two right - angled triangles: one with legs 360 and \( 30 + x \)? No, no. Wait, the horizontal distances: Pierre walks 360m, Dahlia walks 40m. The vertical distances from their starting points to the river edges: Pierre is 30m from his river edge, Dahlia is 50m from her river edge. The river width is \( x \). The two triangles are similar, so \(\frac{360}{40}=\frac{30 + x}{50}\)? Wait, no, actually, the correct proportion is \(\frac{360}{40}=\frac{x + 30}{50}\)? Wait, no, let's re - draw mentally. Pierre starts at a point 30m from his river edge, walks 360m parallel to the river. Dahlia starts at a point 50m from her river edge, walks 40m parallel to the river (opposite direction). The line of sight connects their end points and passes through the two river edges. So the two triangles: one has base 360, height (30 + river width)? No, the river width is \( x \), so the height of Pierre's triangle (from his end point to the line of the river edges) is \( 30 + x \)? No, Dahlia's triangle has base 40, height 50, and Pierre's has base 360, height \( 30 + x \)? Wait, no, the correct similarity ratio: the triangles are similar, so \(\frac{360}{40}=\frac{30 + x}{50}\) is wrong. Wait, actually, the vertical distances from their starting points to the river are 30 and 50, and the horizontal distances they walk are 360 and 40. The river width is \( x \). The two triangles: one with legs 360 and \( 30 + x \), the other with legs 40 and 50? No, that's not right. Wait, the correct proportion is \(\frac{360}{40}=\frac{x + 30}{50}\)? No, let's do it correctly. Let the river width be \( x \). The triangle formed by Pierre's path: horizontal side = 360, vertical side = \( 30 + x \) (from his end point to Dahlia's river edge). The triangle formed by Dahlia's path: horizontal side = 40, vertical side = 50 (from her end point to her river edge). Wait, no, the two triangles are similar, so \(\frac{360}{40}=\frac{30 + x}{50}\) is incorrect. Wait, the correct approach: the two triangles are similar, so \(\frac{360}{40}=\frac{x + 30}{50}\) is wrong. Let's use the correct corresponding sides. The horizontal distance Pierre walks is 360, Dahlia walks 40. The vertical distance from Pierre's starting point to his river edge is 30, from Dahlia's starting point to her river edge is 50. The river width is \( x \). So the ratio of horizontal sides is equal to the ratio of (vertical distance from Pierre's end to Dahlia's river edge) to (vertical distance from Dahlia's end to her river edge). Wait, the vertical distance from Pierre's end to Dahlia's river edge is \( 30 + x \), and from Dahlia's end to her river edge is 50. The horizontal distance for Pierre is 360, for Dahlia is 40. So \(\frac{360}{40}=\frac{30 + x}{50}\). Solving for \( x \):

First, simplify \(\frac{360}{40}=9\). So \( 9=\frac{30 + x}{50}\).

Multiply both sides by 50: \( 9\times50=30 + x \)

\( 450 = 30+x \)

Subtract 30 from both sides: \( x=450 - 30=420 \)? Wait, that can't be right. Wait, I made a mistake in the proportion. Let's re - examine the figure. The two triangles: one has a horizontal leg of 360 and a vertical leg of (river width + 30), the other has a horizontal leg of 40 and a vertical leg of 50? No, the correct corresponding sides: the horizontal distance Pierre walks (360) corresponds to the horizontal distance Dahlia walks (40), and the vertical distance from Pierre's starting point to the far river edge (river width + 30) corresponds to the vertical distance from Dahlia's starting point to her far river edge (50)? No, I think I mixed up. Let's use the basic proportionality for similar triangles. Let the river width be \( x \). The two triangles are similar, so \(\frac{360}{40}=\frac{x + 30}{50}\) is incorrect. Wait, the correct proportion is \(\frac{360}{40}=\frac{30 + x}{50}\) is wrong. Let's start over.

The key is that the two triangles are similar. Let’s denote the river width as \( w \). The triangle with Pierre's movement: base = 360 m, height = \( 30 + w \) (from Pierre's end point to Dahlia's river edge). The triangle with Dahlia's movement: base = 40 m, height = 50 m (from Dahlia's end point to her river edge). Since the triangles are similar, the ratio of bases equals the ratio of heights. So:

\(\frac{360}{40}=\frac{30 + w}{50}\)

Simplify the left - hand side: \(\frac{360}{40}=9\)

So, \( 9=\frac{30 + w}{50}\)

Multiply both sides by 50: \( 9\times50=30 + w \)

\( 450 = 30+w \)

Subtract 30 from both sides: \( w = 450 - 30=420 \)? Wait, that seems too big. Wait, maybe the proportion is \(\frac{360}{40}=\frac{w - 30}{50}\)? No, that doesn't make sense. Wait, maybe I got the vertical sides wrong. Let's look at the figure again. Pierre is 30m from his river edge, Dahlia is 50m from her river edge. The river is between them, so the total vertical distance from Pierre's starting point to Dahlia's starting point is \( 30 + w+ 50 \)? No, no. The two triangles: one is above the river (Pierre's side) with base 360 and height 30, and the other is below the river (Dahlia's side) with base 40 and height 50, and the river width is \( w \). Wait, no, the line of sight connects their end points, passing through the two river edges. So the two triangles are similar: the upper triangle (Pierre's) has base 360 and height 30, the lower triangle (Dahlia's) has base 40 and height 50, and the river width is \( w \). But that can't be, because the heights should be related to the river width. Wait, I think the correct proportion is \(\frac{360}{40}=\frac{30 + w}{50}\) is wrong. Let's use the concept of similar triangles correctly.

Let’s define:

- Let \( w \) be the width of the river.

- The horizontal distance Pierre walks: \( a = 360\) m.

- The horizontal distance Dahlia walks: \( b = 40\) m.

- The vertical distance from Pierre's starting point to his river edge: \( h_1 = 30\) m.

- The vertical distance from Dahlia's starting point to her river edge: \( h_2 = 50\) m.

The two triangles are similar, so \(\frac{a}{b}=\frac{h_1 + w}{h_2}\)

Substitute the values: \(\frac{360}{40}=\frac{30 + w}{50}\)

\( 9=\frac{30 + w}{50}\)

\( 30+w=9\times50 = 450\)

\( w=450 - 30=420\)? Wait, that seems large, but let's check the proportion again. Wait, maybe the proportion is \(\frac{360}{40}=\frac{w - 30}{50}\)? No, that would give a negative number. Wait, maybe I mixed up the vertical sides. Let's think of the two triangles: one has a horizontal leg of 360 and a vertical leg of \( w - 30 \), and the other has a horizontal leg of 40 and a vertical leg of 50. No, that doesn't make sense. Wait, perhaps the correct proportion is \(\frac{360}{40}=\frac{30 + w}{50}\) is correct. Let's verify with the numbers. If \( w = 420 \), then the height for Pierre's triangle is \( 30+420 = 450 \), and \( 450\div50 = 9 \), and \( 360\div40 = 9 \), so the ratios are equal. So the river width is 420 meters.

Answer:

420