Math Problem Statement

Find the missing length indicated. Find PR.

Solution

To solve for PRPR in this triangle problem, we can use the concept of similar triangles. Here’s the analysis step-by-step:

Step-by-Step Solution

  1. Identify Similar Triangles: The two triangles PQK\triangle PQK and PLR\triangle PLR share the angle at PP. Also, they have pairs of corresponding sides that are proportional, making these triangles similar by the AA (Angle-Angle) criterion.

  2. Set Up the Proportion: Since PQKPLR\triangle PQK \sim \triangle PLR, we can set up the following proportion based on the sides: PQPL=PKPR\frac{PQ}{PL} = \frac{PK}{PR}

  3. Substitute Known Values: From the diagram:

    • PQ=5PQ = 5
    • PL=12PL = 12
    • QR=18QR = 18 (Since we are given the full segment PRPR and PLPL is part of it)

    Using these values, rewrite the proportion as: 512=12PR\frac{5}{12} = \frac{12}{PR}

  4. Cross Multiply to Solve for PRPR: [ 5 \cdot PR = 12 \cdot

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Math Problem Analysis

Mathematical Concepts

Similar Triangles
Proportions
Geometry

Formulas

\( \frac{PQ}{PL} = \frac{PK}{PR} \)

Theorems

AA Similarity Criterion

Suitable Grade Level

Grades 9-10