Question

the figure shows two triangles, triangle wzt and triangle xyz, with a common vertex z. the side wt is 8m, ty is 6m, xy is 16m, and xz is labeled as d. angles at w, t, x, y, and z are marked with different symbols indicating congruent angles.

Explanation:

Step1: Identify Similar Triangles

Triangles \( \triangle WTZ \) and \( \triangle XYZ \) are similar (vertical angles and corresponding angles equal). So, their sides are proportional.

Step2: Set Up Proportion

Let the ratio of similarity be based on the sides \( WT = 8 \), \( XY = 16 \), and \( TZ = 6 \), \( XZ = d \) (wait, actually \( TZ = 6 \), and we need to find \( XZ \) or \( d \)? Wait, the sides: \( WT = 8 \), \( XY = 16 \), \( TZ = 6 \), and we need to find \( XZ \) (or \( d \))? Wait, the triangles are similar, so \( \frac{WT}{XY} = \frac{TZ}{XZ} \)? Wait, no, \( \triangle WTZ \sim \triangle XYZ \), so corresponding sides: \( WT \) corresponds to \( XY \), \( TZ \) corresponds to \( YZ \)? Wait, maybe I mixed up. Wait, \( WT = 8 \), \( XY = 16 \), \( TZ = 6 \), and we need to find \( XZ \) (or \( d \))? Wait, the ratio of \( WT \) to \( XY \) is \( 8/16 = 1/2 \). So the ratio of similarity is \( 1/2 \). So \( TZ / YZ = 1/2 \), but wait, \( TZ = 6 \), so \( YZ = 12 \)? Wait, no, maybe the sides are \( WT = 8 \), \( XY = 16 \), \( TZ = 6 \), and we need to find \( XZ \) (or \( d \))? Wait, maybe the triangles are \( \triangle WTZ \) and \( \triangle XYZ \), so \( \frac{WT}{XY} = \frac{TZ}{XZ} \)? Wait, no, let's correct. The vertical angles at \( Z \) are equal, and the other angles are equal (since the triangles are similar by AA similarity). So \( \triangle WTZ \sim \triangle XYZ \) (AA). Therefore, the ratio of corresponding sides is equal. So \( \frac{WT}{XY} = \frac{TZ}{YZ} = \frac{WZ}{XZ} \). Wait, \( WT = 8 \), \( XY = 16 \), so the ratio is \( 8/16 = 1/2 \). So \( TZ / YZ = 1/2 \), but \( TZ = 6 \), so \( YZ = 12 \)? Wait, no, maybe the side \( TZ = 6 \) corresponds to \( XZ \)? Wait, maybe I made a mistake. Wait, the problem is to find \( d \), which is \( XZ \)? Wait, let's re-express. \( \triangle WTZ \sim \triangle XYZ \), so \( \frac{WT}{XY} = \frac{TZ}{XZ} \). Wait, \( WT = 8 \), \( XY = 16 \), \( TZ = 6 \), so \( \frac{8}{16} = \frac{6}{d} \)? Wait, no, that would be \( \frac{8}{16} = \frac{6}{d} \) → \( \frac{1}{2} = \frac{6}{d} \) → \( d = 12 \)? Wait, no, maybe the other way. Wait, \( WT = 8 \), \( XY = 16 \), so the ratio of \( WT \) to \( XY \) is \( 1/2 \), so the ratio of \( TZ \) to \( XZ \) should be \( 1/2 \)? Wait, no, \( TZ \) is a side of the smaller triangle, \( XZ \) is a side of the larger triangle? Wait, no, \( Z \) is the intersection point. So \( WZ \) and \( XZ \) are on the same line, \( TZ \) and \( YZ \) are on the same line. So \( \triangle WTZ \) has sides \( WT = 8 \), \( TZ = 6 \), \( WZ \) (unknown). \( \triangle XYZ \) has sides \( XY = 16 \), \( YZ \) (unknown), \( XZ \) (unknown, which is \( d \)). Since they are similar, \( \frac{WT}{XY} = \frac{TZ}{YZ} = \frac{WZ}{XZ} \). So \( \frac{8}{16} = \frac{6}{YZ} \) → \( YZ = 12 \). But we need to find \( d \), which is \( XZ \). Wait, maybe \( WZ + XZ = WX \), but we don't know \( WX \). Wait, maybe the problem is to find \( d \), which is \( XZ \), and the ratio is \( 8/16 = 6/d \)? Wait, no, that would be \( 8/16 = 6/d \) → \( d = 12 \). Wait, let's check: \( 8/16 = 1/2 \), \( 6/12 = 1/2 \), so that works. So \( d = 12 \).

Wait, maybe the steps are:

Step1: Confirm Similar Triangles

Triangles \( \triangle WTZ \) and \( \triangle XYZ \) are similar by AA similarity (vertical angles at \( Z \) are equal, and the other angles are equal as they are corresponding angles).

Step2: Set Up Proportion

Since \( \triangle WTZ \sim \triangle XYZ \), the ratio of corresponding sides is equal. So:
\[ \frac{WT}{XY} = \frac{TZ}{XZ} \]
Substitute \( WT = 8 \), \( XY = 16 \), \( TZ = 6 \):
\[ \frac{8}{16} = \frac{6}{d} \]

Step3: Solve for \( d \)

Simplify \( \frac{8}{16} = \frac{1}{2} \), so:
\[ \frac{1}{2} = \frac{6}{d} \]
Cross - multiply:
\[ d = 6 \times 2 = 12 \]

Answer:

\( d = 12 \)