Math Problem Statement

What is the length of side x in Triangle 2, which is a scale drawing of Triangle 1?

Solution

In the given problem, Triangle 2 is a scale drawing of Triangle 1, meaning the two triangles are similar. Similar triangles have proportional sides. Let's use this information to find the length of side xx in Triangle 2.

Information from the image:

  • Triangle 1:

    • Short leg = 12 cm
    • Long leg = 16 cm
    • Hypotenuse = 20 cm
  • Triangle 2:

    • Short leg = 30 cm
    • Long leg = xx
    • Hypotenuse = 50 cm

Since the triangles are similar, the ratio of corresponding sides is constant. The ratio between the hypotenuses is:

5020=52\frac{50}{20} = \frac{5}{2}

So, the sides of Triangle 2 are scaled by a factor of 52\frac{5}{2} compared to Triangle 1. We can apply this scale factor to the long leg (16 cm) of Triangle 1 to find xx:

x=16×52=16×2.5=40cmx = 16 \times \frac{5}{2} = 16 \times 2.5 = 40 \, \text{cm}

Thus, the length of side xx is 40 cm.

Would you like further details or have any other questions? Here are some related questions you could explore:

  1. How can we verify that the triangles are indeed similar?
  2. How would you find the area of both triangles?
  3. What is the ratio of the areas of Triangle 1 and Triangle 2?
  4. If the short leg of Triangle 1 was increased, how would that affect the dimensions of Triangle 2?
  5. How does the Pythagorean theorem apply to both triangles?

Tip: When working with similar triangles, remember that all corresponding angles are equal, and corresponding sides are proportional.

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Math Problem Analysis

Mathematical Concepts

Geometry
Proportions
Similar Triangles

Formulas

Proportion of similar triangles: (corresponding side 1)/(corresponding side 2) = (scale factor)

Theorems

Pythagorean Theorem
Similarity Theorem

Suitable Grade Level

Grades 7-8