Triangle Classification Using Side Lengths
We are given a triangle \(\Delta\text{ABC}\) with the following side lengths:
- \(\text{AB} = 16\text{cm}\)
- \(\text{BC} = 63\text{cm}\)
- \(\text{AC} = 65\text{cm}\)
Pythagorean Theorem Converse Check
To determine if the triangle is right-angled, acute, or obtuse, we use the converse of the Pythagorean theorem. We need to compare the square of the longest side with the sum of the squares of the other two sides.
Calculating Side Squares
First, calculate the squares of each side length:
- Square of side AB: \(\text{AB}^2 = (16\text{cm})^2 = 256\text{cm}^2\)
- Square of side BC: \(\text{BC}^2 = (63\text{cm})^2 = 3969\text{cm}^2\)
- Square of side AC (the longest side): \(\text{AC}^2 = (65\text{cm})^2 = 4225\text{cm}^2\)
Verifying the Theorem Condition
Next, sum the squares of the two shorter sides:
\(\text{AB}^2 + \text{BC}^2 = 256\text{cm}^2 + 3969\text{cm}^2 = 4225\text{cm}^2\)
Compare this sum to the square of the longest side:
We found that \(\text{AB}^2 + \text{BC}^2 = 4225\text{cm}^2\), and \(\text{AC}^2 = 4225\text{cm}^2\). Therefore, \(\text{AB}^2 + \text{BC}^2 = \text{AC}^2\).
Conclusion on Triangle Type
According to the converse of the Pythagorean theorem, if \(a^2 + b^2 = c^2\), the triangle is a right-angled triangle. Since the condition is met, triangle \(\Delta\text{ABC}\) is a Right angled Triangle.
Similar Questions
- It is given that, \(\Delta ABC \sim \Delta EDF\) such that \(AB = 5 \text{ cm}\), \(AC = 7.5 \text{ cm}\), \(DF = 15 \text{ cm}\) and \(ED = 12.5 \text{ cm}\). The sum of the remaining sides of the triangles is:
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- Triangles $PQR$ and $ABC$ are similar triangles.
Triangle $ABC$ have the following sides.
$AB = 36\text{ cm}$
$AC = 20\text{ cm}$
$BC = 48\text{ cm}$
Triangle $PQR$ have the following sides:
$PQ = 9\text{ cm}$
$PR = 5\text{ cm}$
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Important Questions from Triangles
Among the following options, which are NOT sides of a triangle?
In a Δ ABC, if ∠A = 120° and AB = AC, then the values of ∠B and ∠C are respectively:
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A man goes 24 m towards east and then 10 m towards north. How far is he away from his initial position?
