Math Problem Statement

Triangle FGH is similar to triangle IJK. Find IJ. Round your answer to the nearest tenth if necessary. Figures are not drawn to scale.

Solution

In this problem, we have two similar triangles, FGH\triangle FGH and IJK\triangle IJK. The side lengths of these triangles are given, and we are asked to find the length of IJIJ (denoted as xx in the diagram).

Step-by-Step Solution

  1. Identify the corresponding sides: Since the triangles are similar, their corresponding sides are proportional. This gives us: FGIJ=GHJK\frac{FG}{IJ} = \frac{GH}{JK}

  2. Assign known values to the ratios:

    • FG=23FG = 23
    • GH=17GH = 17
    • IJ=xIJ = x (unknown)
    • JK=58JK = 58

    Using the proportion: 23x=1758\frac{23}{x} = \frac{17}{58}

  3. Solve for xx: Cross-multiply to solve for xx: 23×58=17×x23 \times 58 = 17 \times x 1334=17x1334 = 17x x=133417x = \frac{1334}{17} x78.5x \approx 78.5

Answer

The length of IJIJ is approximately 78.5.

Would you like more details on the solution or have any questions?

Here are some related questions for further understanding:

  1. How do you identify corresponding sides in similar triangles?
  2. What is the cross-multiplication method, and how does it work in proportions?
  3. Why are similar triangles useful in solving for unknown lengths?
  4. How do you round numbers to the nearest tenth, and why is it necessary here?
  5. What other real-life applications use triangle similarity and proportional reasoning?

Tip: Always double-check which sides correspond in similar triangles, as this ensures accurate setting up of the proportion.

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Math Problem Analysis

Mathematical Concepts

Geometry
Similarity of Triangles
Proportions

Formulas

\( \frac{FG}{IJ} = \frac{GH}{JK} \)

Theorems

Similarity Theorem for Triangles

Suitable Grade Level

Grades 8-10