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 with a smaller triangle within it, and both triangles share the same height. From the diagram:
- , since is extended by .
- and are marked as proportional segments.
Solving the Problem
The triangles and are similar (AA similarity), so their corresponding side ratios are equal. This gives:
1. Ratio of Similar Triangles
From the height proportions, let’s denote the ratio . Thus:
2. Set Proportions
The ratio 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!Ask a new question for Free
By Image
Drop file here or Click Here to upload
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