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Complete the Triangle Similarity Statement for △TUV and △QSR
Mathematics
Grade 10 (Junior High School)
Question Content
Finish the similarity statement below: (UPPER CASE LETTERS PLEASE). Given triangle TUV with side lengths TU=70, UV=84, VT=42, and triangle QSR with side lengths QS=15, SR=30, RQ=25. Complete the statement: △TUV ~ △____
Correct Answer
△SRQ
Detailed Solution Steps
1
Step 1: Calculate the ratio of corresponding sides of the two triangles. First, sort the side lengths of each triangle from shortest to longest: For △TUV: 42, 70, 84; For △QSR: 15, 25, 30.
2
Step 2: Find the scale factor by dividing the sides of △TUV by the corresponding sides of △QSR: 42/15 = 2.8, 70/25 = 2.8, 84/30 = 2.8. All ratios are equal, confirming similarity.
3
Step 3: Match the sides in order: The shortest side of △TUV (VT=42) corresponds to the shortest side of △QSR (QS=15); the middle side of △TUV (TU=70) corresponds to the middle side of △QSR (RQ=25); the longest side of △TUV (UV=84) corresponds to the longest side of △QSR (SR=30).
4
Step 4: Map the vertices based on side correspondence: V ↔ Q, T ↔ R, U ↔ S. Rearrange to match the order of △TUV: T corresponds to S, U corresponds to R, V corresponds to Q, so △TUV ~ △SRQ.
Knowledge Points Involved
1
Triangle Similarity (SSS Criterion)
The Side-Side-Side (SSS) similarity criterion states that if the ratios of the lengths of all three pairs of corresponding sides of two triangles are equal, then the two triangles are similar. This is used to confirm that two triangles have the same shape, even if their sizes are different.
2
Corresponding Sides in Similar Triangles
In similar triangles, corresponding sides are the sides that are in the same relative position in each triangle, and their lengths have a constant ratio called the scale factor. Matching corresponding sides correctly is essential for writing accurate similarity statements.
3
Similarity Statement Notation
When writing a similarity statement (△ABC ~ △DEF), the order of the vertices indicates the corresponding angles and sides. Vertex A corresponds to D, B to E, and C to F, meaning ∠A≅∠D, ∠B≅∠E, ∠C≅∠F, and AB/DE = BC/EF = AC/DF.
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