In a right-angled triangle, we have three sides of lengths namely hypotenuse H cm, base B cm and height T cm. Then which of the following ratios is the largest ?
\(\frac{H}{T}\)
In a right-angled triangle the hypotenuse lies opposite the right angle, which is the largest angle, so the hypotenuse is the longest side. That gives \(H > B\) and \(H > T\), and this single fact orders all four ratios.
The two ratios with \(H\) in the denominator are each less than 1, since \(T < H\) gives \(\frac{T}{H} < 1\) and \(B < H\) gives \(\frac{B}{H} < 1\).
The ratio \(\frac{H}{T}\) is greater than 1, because its numerator is the longest side and its denominator is not. So it already exceeds both of the ratios above.
The two remaining ratios share the denominator \(T\), so they are ordered by their numerators alone. Since \(H > B\), it follows that \(\frac{H}{T} > \frac{B}{T}\).
So \(\frac{H}{T}\) beats every other option, and the conclusion holds for any right-angled triangle, whatever the shape — a check with legs 3 and 4 and hypotenuse 5 gives the values 1.67, 1.33, 0.6 and 0.8, in the same order.
Hence the largest ratio is \(\frac{H}{T}\).
Important Questions from Triangles
In a Δ ABC, if ∠A = 120° and AB = AC, then the values of ∠B and ∠C are respectively:
The product of the perimeter of a triangle, the radius of its in‐circle, and a number gives the area of the triangle. The number is
In ∆ABC, 2∠A = 3∠B = 6∠C. What is the value of the largest angle among these three angles?
The longest side in a right-angled triangle is called :
The sum of two sides of a triangle with the third side is
