Math Problem Statement

In the following figure , ST II QR , point S divides PQ in the ratio 4:5. If PT=1.6 cm , what is the length of PR ?

Solution

To solve this problem, we are given the following information:

  • STQRST \parallel QR
  • Point SS divides PQPQ in the ratio 4:5.
  • The length PT=1.6cmPT = 1.6 \, \text{cm}.
  • We are asked to find the length of PRPR.

Step-by-Step Solution:

  1. Since STQRST \parallel QR, by using the Basic Proportionality Theorem (also known as Thales' theorem), the line segment SS divides PQPQ in the same ratio that it divides PRPR.

  2. Let the total length of PRPR be denoted by xcmx \, \text{cm}.

  3. Point SS divides PQPQ in the ratio 4:5, meaning that:

    • The length of PS=44+5×PQ=49×PQPS = \frac{4}{4+5} \times PQ = \frac{4}{9} \times PQ.
    • The length of SQ=59×PQSQ = \frac{5}{9} \times PQ.
  4. We are also given that PT=1.6cmPT = 1.6 \, \text{cm}, and since STQRST \parallel QR, the entire segment PTPT corresponds to PRPR in the same ratio. This means: PTPR=PSPQ\frac{PT}{PR} = \frac{PS}{PQ}

  5. Using the ratio 49\frac{4}{9}, we can set up the proportion: 1.6x=49\frac{1.6}{x} = \frac{4}{9}

  6. Solving for xx: x=9×1.64=3.6cmx = \frac{9 \times 1.6}{4} = 3.6 \, \text{cm}

Thus, the length of PRPR is 3.6 cm.

Would you like further details or have any questions?

Here are 5 related questions to consider:

  1. How does the Basic Proportionality Theorem apply in various types of triangles?
  2. Can the theorem be used in 3D geometry, and if so, how?
  3. How does the division ratio change if STST and QRQR are not parallel?
  4. What would happen if the ratio of division was reversed (5:4)?
  5. How would the result change if you were asked for PQPQ instead of PRPR?

Tip: When working with ratios, always set up proportions carefully to ensure consistency between the terms being compared.

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Math Problem Analysis

Mathematical Concepts

Proportions
Ratios
Geometry

Formulas

Ratio formula: PS/PQ = PT/PR
Proportion formula: PT/PR = 4/9

Theorems

Basic Proportionality Theorem (Thales' Theorem)

Suitable Grade Level

Grades 8-10