Question**: A triangular plot of land has sides measuring 7 meters, 24 meters, and 25 meters. Is the triangle a right triangle, and if so, what is its area? - Decision Point
Is a Triangle with Sides 7 m, 24 m, and 25 m a Right Triangle? Calculate Its Area
Is a Triangle with Sides 7 m, 24 m, and 25 m a Right Triangle? Calculate Its Area
If you’ve come across a triangular plot of land with side lengths 7 meters, 24 meters, and 25 meters and wondered whether it forms a right triangle, you’re in the right place. Not only can we determine this with a simple math test, but we’ll also calculate the area of this promising plot of land.
Understanding the Context
How to Check if a Triangle is a Right Triangle
A triangle is a right triangle if the sides satisfy the Pythagorean Theorem:
a² + b² = c², where c is the longest side (hypotenuse).
Given sides:
- 7 m
- 24 m
- 25 m
First, identify the largest side:
25 m is the longest side → likely the hypotenuse.
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Key Insights
Check:
7² + 24² = ? 25²
7² = 49
24² = 576
25² = 625
Now calculate:
49 + 576 = 625
✅ 49 + 576 = 625, so:
7² + 24² = 25²
This confirms the triangle is a right triangle.
What Type of Right Triangle is It?
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This triangle has sides that are a multiple of the classic 3-4-5 triangle scaled by 7:
- 7 → 3×7 = 21? No — wait, actually 7-24-25 is a well-known Pythagorean triple.
Indeed, 7² + 24² = 25² proves it’s a valid right triangle, though not one of the smallest well-known ones.
How to Calculate the Area of the Triangle
For a right triangle, the area is simply:
\[
\ ext{Area} = \frac{1}{2} \ imes \ ext{base} \ imes \ ext{height}
\]
Using 7 m and 24 m as the legs (since they’re perpendicular):
\[
\ ext{Area} = \frac{1}{2} \ imes 7 \ imes 24 = \frac{1}{2} \ imes 168 = 84 \ ext{ m²}
\]
✅ Area = 84 square meters
Why This Matters for Landowners
Knowing whether a plot is a right triangle helps in:
- Calculating exact land area
- Planning structures or fencing
- Optimizing space usage in construction or landscaping
