Math Problem Statement

In the trapezoid below, TU is parallel to QR. If QS = 35, TS = 15, QR = 28, and SR = 21, find the length of SU.

Solution

The problem describes a trapezoid TURSTURS, where TUQRTU \parallel QR, and provides several lengths. We need to find the length of segment SUSU.

Given Information:

  • QS=35QS = 35
  • TS=15TS = 15
  • QR=28QR = 28
  • SR=21SR = 21

We are asked to find the length of SUSU.

Steps:

  1. Use the Trapezoid Proportionality Theorem: Since TUQRTU \parallel QR, we can apply the trapezoid midsegment theorem or use the property that triangles formed by diagonals of the trapezoid are proportional.

  2. Identify Similar Triangles: Triangles TUSTUS and QRSQRS are similar by the AA similarity criterion because both share angle SS and the angles formed by the parallel lines TUTU and QRQR are congruent.

  3. Set Up Proportion: Since the triangles are similar, the sides are proportional. Thus, the ratio of corresponding sides is: SUSR=TSQR\frac{SU}{SR} = \frac{TS}{QR}

  4. Substitute Known Values:

    • SR=21SR = 21
    • TS=15TS = 15
    • QR=28QR = 28

    SU21=1528\frac{SU}{21} = \frac{15}{28}

  5. Solve for SUSU: SU=1528×21=15×2128=31528=11.25SU = \frac{15}{28} \times 21 = \frac{15 \times 21}{28} = \frac{315}{28} = 11.25

Thus, the length of SUSU is 11.25 units.

Would you like more details or clarifications? Here are five related questions you might find helpful:

  1. How can we use properties of trapezoids in other types of geometric problems?
  2. What is the trapezoid midsegment theorem, and when is it applicable?
  3. Can the similarity of triangles be applied in more complex shapes?
  4. How does proportional reasoning simplify calculations in geometry?
  5. What other geometric figures share properties similar to trapezoids?

Tip: Whenever working with parallel lines in polygons, check for similar triangles to simplify your calculations!

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Math Problem Analysis

Mathematical Concepts

Geometry
Similar Triangles
Proportions

Formulas

(SU / SR) = (TS / QR)

Theorems

Trapezoid Proportionality Theorem
AA Similarity Theorem

Suitable Grade Level

Grades 9-11