Math Problem Statement

The problem involves a triangle \( \triangle VWT \) with a smaller triangle \( \triangle XWT \) within it, and both triangles share the same height. From the diagram, 1. \( VW = 24 \), 2. \( WT = 72 \), 3. \( VT = 17 + UT \), since \( VU \) is extended by \( UT \). 4. \( WX \) and \( XT \) are marked as proportional segments. Find the lengths of segments \( XT \), \( UT \), and \( VT \).

Solution

The problem involves a triangle VWT\triangle VWT with a smaller triangle XWT\triangle XWT within it, and both triangles share the same height. From the diagram:

  1. VW=24VW = 24
  2. WT=72WT = 72
  3. VT=17+UTVT = 17 + UT, since VUVU is extended by UTUT.
  4. WXWX and XTXT are marked as proportional segments.

Solving the Problem

The triangles VWT\triangle VWT and XWT\triangle XWT are similar (AA similarity), so their corresponding side ratios are equal. This gives:

WXVW=XTWT=VTVT.\frac{WX}{VW} = \frac{XT}{WT} = \frac{VT}{VT}.

1. Ratio of Similar Triangles

From the height proportions, let’s denote the ratio k=WXVWk = \frac{WX}{VW}. Thus: k=XTWT.k = \frac{XT}{WT}.

2. Set Proportions

The ratio kk is determined by the segment heights:

Since $$\triangle VWT$$ and $$\triangle XWT$$ are similar, we can write proportional relationships between corresponding sides: \[ \frac{WX}{VW} = \frac{XT}{WT} = \frac{VT}{UT}.$$ ### Proportions and Known Values From the diagram: 1. $$WX = 24$$, $$VW = 72$$, $$WT = 72$$. 2. Let $$XT$$, $$UT$$, and $$VT$$ be unknowns, which we are solving for. The proportions yield: $$\frac{WX}{VW} = \frac{XT}{WT}.$$ Substituting $$WX = 24$$, $$VW = 72$$, and $$WT = 72$$: $$\frac{24}{72} = \frac{XT}{72}.$$ Simplify the fraction: $$\frac{1}{3} = \frac{XT}{72}.$$ Solve for $$XT$$: $$XT = \frac{72}{3} = 24.$$ #### Solve for $$UT$$ From the similarity of triangles, the remaining length $$UT$$ is proportional. Using the height ratio $$\frac{WX}{VW} = \frac{1}{3}$$, we can find: $$UT = \frac{1}{3} \times VT.$$ Since $$VT = 17 + UT$$, substitute $$UT = \frac{1}{3} \times VT$$ into the equation: $$VT = 17 + \frac{1}{3} VT.$$ Multiply through by 3 to eliminate the fraction: $$3VT = 51 + VT.$$ Solve for $$VT$$: $$2VT = 51 \implies VT = \frac{51}{2} = 25.5.$$ #### Find $$UT$$ Substitute $$VT = 25.5$$ into $$UT = \frac{1}{3} \times VT$$: $$UT = \frac{1}{3} \times 25.5 = 8.5.$$ ### Final Answers - $$XT = 24$$ - $$UT = 8.5$$ - $$VT = 25.5$$ Would you like me to clarify any steps or elaborate further? --- ### 5 Related Questions 1. How do you verify the similarity of triangles? 2. How can you generalize the proportions for different triangle sizes? 3. What happens if the given ratio of segments changes? 4. How can this concept be applied to real-world geometric problems? 5. What alternative methods can you use to solve for missing segments? ### Tip Always confirm the similarity of triangles before applying proportions. Look for congruent angles or proportional side ratios!

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Math Problem Analysis

Mathematical Concepts

Triangle Similarity
Proportions
Geometry

Formulas

\frac{WX}{VW} = \frac{XT}{WT} = \frac{VT}{UT}

Theorems

AA Similarity Theorem

Suitable Grade Level

Grades 8-10