Question

$\overline{wx} \parallel \overline{vy}$. find $wz$.
(there is a geometric figure with points x, y, z, w, v. xy and yz are on the base, yz = 4, wz is a vertical side with wv = 2, xz total length is 8. we need to find wz.)
$wz = \square$
submit

Explanation:

Step1: Identify Similar Triangles

Since \( \overline{WX} \parallel \overline{VY} \), by the Basic Proportionality Theorem (Thales' theorem), triangles \( \triangle WXZ \) and \( \triangle VYZ \) are similar. So the ratios of corresponding sides are equal.

Step2: Set Up Proportion

Let \( WZ = x \), then \( VZ = x - 2 \) (since \( WV = 2 \)). The ratio of \( XZ \) to \( YZ \) should equal the ratio of \( WZ \) to \( VZ \). We know \( XZ = 8 \), \( YZ = 4 \), so:

$$ \frac{XZ}{YZ}=\frac{WZ}{VZ} \implies \frac{8}{4}=\frac{x}{x - 2} $$

Step3: Solve the Proportion

Simplify \( \frac{8}{4}=2 \), so the equation becomes:

$$ 2=\frac{x}{x - 2} $$

Multiply both sides by \( x - 2 \):

$$ 2(x - 2)=x $$

Expand the left side:

$$ 2x - 4 = x $$

Subtract \( x \) from both sides:

$$ x - 4 = 0 \implies x = 4 $$

Wait, that seems incorrect. Wait, maybe the ratio is \( \frac{XZ}{YZ}=\frac{WZ}{WV + VZ} \)? Wait, no, let's re - examine the diagram. Wait, \( XZ = 8 \), \( YZ = 4 \), so \( XY=XZ - YZ = 8 - 4 = 4 \). Wait, maybe the correct proportion is from the Basic Proportionality Theorem: if a line is drawn parallel to one side of a triangle, intersecting the other two sides, then it divides those sides proportionally. So in \( \triangle WXZ \), \( VY \parallel WX \), so \( \frac{YZ}{XZ}=\frac{VZ}{WZ} \). Let \( WZ = x \), \( VZ=x - 2 \), \( YZ = 4 \), \( XZ = 8 \). Then:

$$ \frac{4}{8}=\frac{x - 2}{x} $$

Cross - multiply:

$$ 4x=8(x - 2) $$

Expand:

$$ 4x = 8x-16 $$

Subtract \( 4x \) from both sides:

$$ 0 = 4x - 16 $$

Add 16 to both sides:

$$ 4x=16 $$

Divide by 4:

$$ x = 4 $$

Wait, but \( WV = 2 \), \( VZ=x - 2=2 \), so \( WZ=WV + VZ=2 + 2 = 4 \)? Wait, maybe the initial proportion was wrong. Let's use the correct similar triangles. Since \( VY\parallel WX \), \( \triangle VYZ\sim\triangle WXZ \). So the ratio of corresponding sides: \( \frac{YZ}{XZ}=\frac{VZ}{WZ} \). \( YZ = 4 \), \( XZ = 8 \), so the ratio of similarity is \( \frac{4}{8}=\frac{1}{2} \). So \( VZ=\frac{1}{2}WZ \). Let \( WZ = x \), then \( VZ=x - 2 \) (because \( WV = 2 \)). So \( x - 2=\frac{1}{2}x \). Multiply both sides by 2: \( 2x-4=x \), so \( x = 4 \). So \( WZ = 4 \).

Answer:

\( 4 \)