Math Problem Statement

Find GP in the given triangle with the given segment lengths.

Solution

The problem involves finding the length of segment GPGP in a triangle with parallel segments. We are given the lengths of FG=3.3FG = 3.3, GH=4.5GH = 4.5, and PQ=2.7PQ = 2.7. Since PQPQ is parallel to FHFH, triangles FGHFGH and GPQGPQ are similar by AA similarity.

To find GPGP, we can set up a proportion using the corresponding sides of the similar triangles:

GPFG=PQGH\frac{GP}{FG} = \frac{PQ}{GH}

Substituting the known values:

GP3.3=2.74.5\frac{GP}{3.3} = \frac{2.7}{4.5}

Simplify the right side:

GP3.3=35\frac{GP}{3.3} = \frac{3}{5}

Now, solve for GPGP:

GP=3.3×35=1.982.0GP = 3.3 \times \frac{3}{5} = 1.98 \approx 2.0

The closest option is 2.1, so that would be the correct choice.

Would you like further details or have any questions?

Related Questions:

  1. What property allows triangles to be similar when they have parallel sides?
  2. How can we verify that the ratio between the corresponding sides is correct?
  3. What would happen if the triangles were not similar? Could we still solve the problem?
  4. How do you simplify a fraction like 2.74.5\frac{2.7}{4.5}?
  5. Why is it important to round to the nearest option in this case?

Tip:

Always check if two triangles share a common angle and if their corresponding sides are proportional to confirm similarity.

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Math Problem Analysis

Mathematical Concepts

Similar triangles
Proportions
Geometry

Formulas

GP / FG = PQ / GH
GP = FG * (PQ / GH)

Theorems

AA Similarity Theorem

Suitable Grade Level

Grades 8-10