Math Problem Statement

Find TU in the given triangle using the Triangle Proportionality Theorem. The triangle has QR = 15, RS = 30, ST = 38, and TU is unknown.

Solution

The problem involves the Triangle Proportionality Theorem. According to this theorem, if a line parallel to one side of a triangle intersects the other two sides, it divides them proportionally.

Here’s the given data:

  1. QR=15QR = 15, RS=30RS = 30
  2. ST=38ST = 38, TU=xTU = x (to be found).

Step-by-step solution:

  1. Using the Triangle Proportionality Theorem, the ratios of the divided sides are equal: QRRS=STTU\frac{QR}{RS} = \frac{ST}{TU}

  2. Substitute the known values: 1530=38x\frac{15}{30} = \frac{38}{x}

  3. Simplify the first fraction: 12=38x\frac{1}{2} = \frac{38}{x}

  4. Cross-multiply: 1x=2381 \cdot x = 2 \cdot 38

  5. Solve for xx: x=76x = 76

Thus, TU=76TU = 76.

Would you like additional details or explanations?

Related Questions:

  1. What is the Triangle Proportionality Theorem and how does it work?
  2. How can we verify the ratios in triangle proportionality problems?
  3. What are some real-life applications of proportionality in geometry?
  4. What happens when the dividing line is not parallel to one side of the triangle?
  5. How does the Triangle Proportionality Theorem relate to similar triangles?

Tip:

When solving proportionality problems in triangles, always verify that the line dividing the two sides is parallel to the third side. This ensures the theorem is applicable.

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Math Problem Analysis

Mathematical Concepts

Geometry
Proportionality

Formulas

QR/RS = ST/TU

Theorems

Triangle Proportionality Theorem

Suitable Grade Level

Grades 8-10