Question

modeling real life you can measure the width of the lake using a surveying technique, as shown in the diagram. find the width of the lake, ( wx ). ( wx = square ) m

Explanation:

Step1: Identify similar triangles

Triangles \( \triangle VWX \) and \( \triangle ZYX \) are similar (by AA similarity, as \( \angle VWX = \angle ZYX = 90^\circ \) and \( \angle VXW = \angle ZXY \) (vertical angles)). So, the ratios of corresponding sides are equal.
\[ \frac{VW}{ZY} = \frac{WX}{YX} \]

Step2: Substitute known values

We know \( VW = 104 \, \text{m} \), \( ZY = 8 \, \text{m} \), \( YX = 6 \, \text{m} \)? Wait, no, wait. Wait, \( ZX = 6 \, \text{m} \), \( YZ = 8 \, \text{m} \)? Wait, no, let's re - check. Wait, the sides: \( VW = 104 \), \( ZY = 8 \), \( XZ = 6 \), and we need to find \( WX \). Wait, actually, the correct correspondence: \( \triangle VWX \sim \triangle ZYX \), so \( \frac{VW}{ZY}=\frac{WX}{ZX} \)? Wait, no, maybe I mixed up. Wait, \( \angle W = \angle Z = 90^\circ \), \( \angle VXW=\angle YXZ \) (vertical angles). So sides: \( VW \) corresponds to \( ZY \), \( WX \) corresponds to \( ZX \)? Wait, no, \( YZ = 8 \), \( ZX = 6 \), \( VW = 104 \), and we need \( WX \). Let's set up the proportion correctly. Since \( \triangle VWX \sim \triangle ZYX \), then \( \frac{VW}{ZY}=\frac{WX}{YX} \)? Wait, no, \( YX \) is 8? Wait, the diagram: \( ZY = 8 \, \text{m} \), \( XZ = 6 \, \text{m} \), \( VW = 104 \, \text{m} \). So the correct proportion is \( \frac{VW}{ZY}=\frac{WX}{XZ} \)? Wait, no, let's use the similarity of triangles. If \( \triangle VWX \sim \triangle ZYX \), then \( \frac{VW}{ZY}=\frac{WX}{YX} \)? Wait, I think I made a mistake. Let's do it again. Let's denote \( WX = x \). The two right - angled triangles: \( \triangle VWX \) (right - angled at \( W \)) and \( \triangle ZYX \) (right - angled at \( Z \)). The angles at \( X \) are vertical angles, so they are equal. So, by AA similarity, \( \triangle VWX \sim \triangle ZYX \). Therefore, the ratio of corresponding sides: \( \frac{VW}{ZY}=\frac{WX}{YX} \). Wait, \( ZY = 8 \), \( YX = 6 \)? No, \( ZX = 6 \), \( YZ = 8 \). Wait, maybe the sides are \( VW = 104 \), \( ZY = 8 \), \( XZ = 6 \), and \( WX = x \). Wait, no, the correct proportion is \( \frac{VW}{ZY}=\frac{WX}{XZ} \)? Wait, no, let's use the formula for similar triangles. Let's let \( WX = x \). Then:

Since \( \triangle VWX \sim \triangle ZYX \), we have \( \frac{VW}{ZY}=\frac{WX}{ZX} \)

Substitute \( VW = 104 \), \( ZY = 8 \), \( ZX = 6 \):

\[ \frac{104}{8}=\frac{x}{6} \]

Wait, no, that can't be. Wait, maybe the other way. Wait, \( ZY = 8 \), \( YX = 6 \)? No, the diagram: \( Z \) to \( Y \) is 8m, \( Z \) to \( X \) is 6m, \( V \) to \( W \) is 104m. So the correct proportion is \( \frac{VW}{ZY}=\frac{WX}{YX} \), where \( YX = 6 \), \( ZY = 8 \), \( VW = 104 \). Wait, no, I think I messed up the correspondence. Let's use the correct similarity ratio. Let's assume that \( \triangle VWX \sim \triangle ZYX \), so \( \frac{VW}{ZY}=\frac{WX}{XZ} \). Wait, \( XZ = 6 \), \( ZY = 8 \), \( VW = 104 \). Then:

\[ \frac{104}{8}=\frac{x}{6} \]

Wait, solving for \( x \): \( x=\frac{104\times6}{8} \)

Wait, no, \( 104\div8 = 13 \), then \( 13\times6 = 78 \)? Wait, no, wait, maybe the sides are \( ZY = 8 \), \( XZ = 6 \), \( VW = 104 \), and the ratio is \( \frac{VW}{ZY}=\frac{WX}{XZ} \). So:

\[ \frac{104}{8}=\frac{WX}{6} \]

Multiply both sides by 6: \( WX=\frac{104\times6}{8} \)

Step3: Calculate the value

First, simplify \( \frac{104}{8}=13 \). Then \( 13\times6 = 78 \)? Wait, no, wait, 104 divided by 8 is 13, 13 times 6 is 78? Wait, no, wait, maybe the proportion is \( \frac{VW}{ZY}=\frac{WX}{YX} \), where \( YX = 8 \) and \( XZ = 6 \)? No, I think I had the correspondence wrong. Let's start over.

The two triangles: \( \triangle VWX \) (right - angled at \( W \)) and \( \triangle ZYX \) (right - angled at \( Z \)). The angles at \( X \) are vertical angles, so they are equal. So, the similarity ratio is based on the legs. So, \( \frac{VW}{ZY}=\frac{WX}{ZX} \). Wait, \( VW = 104 \), \( ZY = 8 \), \( ZX = 6 \). Then:

\[ WX=\frac{VW\times ZX}{ZY}=\frac{104\times6}{8} \]

Calculate \( 104\times6 = 624 \), then \( 624\div8 = 78 \). Wait, no, 104 divided by 8 is 13, 13 times 6 is 78? Wait, 8 times 13 is 104, so 104/8 = 13, then 13*6 = 78.

Wait, but let's check the proportion again. If \( \triangle VWX \sim \triangle ZYX \), then \( \frac{VW}{ZY}=\frac{WX}{YX} \), where \( YX = 8 \) and \( ZX = 6 \)? No, I think the correct correspondence is \( VW \) corresponds to \( ZY \), \( WX \) corresponds to \( ZX \), and \( VX \) corresponds to \( YX \). So the proportion is \( \frac{VW}{ZY}=\frac{WX}{ZX} \). So with \( VW = 104 \), \( ZY = 8 \), \( ZX = 6 \), we get \( WX=\frac{104\times6}{8}=78 \). Wait, no, 104*6 = 624, 624/8 = 78.

Answer:

\( 78 \)