A triangle has sides in the ratio 5:12:13. If the shortest side is 10 cm, what is the area of the triangle? - Decision Point
Using the 5:12:13 Triangle Ratio to Find Area – A Simple Geometry Guide
Using the 5:12:13 Triangle Ratio to Find Area – A Simple Geometry Guide
A triangle with sides in the ratio 5:12:13 is a classic example of a right triangle. If the shortest side measures 10 cm and follows this ratio, it serves as a perfect starting point to explore triangle properties—especially area calculation.
In a 5:12:13 triangle, the sides are proportional to 5k, 12k, and 13k, where k is a scaling factor. Since the shortest side is 10 cm and corresponds to 5k, we solve:
Understanding the Context
\[
5k = 10 \implies k = 2
\]
Using this factor:
- Second side = 12k = 12 × 2 = 24 cm
- Longest side = 13k = 13 × 2 = 26 cm
Because 5² + 12² = 25 + 144 = 169 = 13², this triangle is a right triangle, with the right angle between the sides of 10 cm and 24 cm.
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Key Insights
Calculating the Area
For a right triangle, area is simply:
\[
\ ext{Area} = \frac{1}{2} \ imes \ ext{base} \ imes \ ext{height}
\]
Here, base = 10 cm and height = 24 cm:
\[
\ ext{Area} = \frac{1}{2} \ imes 10 \ imes 24 = \frac{1}{2} \ imes 240 = 120~\ ext{cm}^2
\]
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So, the area of the triangle with sides in the ratio 5:12:13 and shortest side 10 cm is 120 square centimeters.
Why This Ratio Matters
The 5:12:13 triangle is famous because it generates one of the smallest whole-number-sided Pythagorean triples. This makes it ideal for geometry lessons, construction, and understanding proportional relationships in right triangles.
If you’re studying triangles, recognizing this ratio helps simplify calculations—especially area and perimeter—without complex formulas.
In summary:
- Ratio: 5:12:13
- Shortest side: 10 cm → k = 2
- Other sides: 24 cm and 26 cm
- Area: 120 cm²
Mastering these basic triangles builds a strong foundation in geometry!
