If the sides of a triangle are in the ratio `3 : 1 1/4 :3 1/4`. then the triangle is

Options

A. a) Right triangle
B. b) Obtuse triangle
C. c) Equiangular triangle
D. d) Acute triangle

Answer

To determine the type of triangle formed by the sides in the ratio of \(3 : 1 \frac{1}{4} : 3 \frac{1}{4}\), we will follow these steps:

### Step 1: Convert the Ratios to Improper Fractions
First, we need to convert the mixed numbers into improper fractions for easier calculation.

- \(1 \frac{1}{4} = \frac{5}{4}\)
- \(3 \frac{1}{4} = \frac{13}{4}\)

So, the sides of the triangle can be expressed as:
- Side 1: \(3\)
- Side 2: \(\frac{5}{4}\)
- Side 3: \(\frac{13}{4}\)

### Step 2: Write the Ratios as a Fraction
Now we can express the sides in a common format:
- The ratio becomes \(3 : \frac{5}{4} : \frac{13}{4}\).

### Step 3: Eliminate the Fractions
To eliminate the fractions, we can multiply all parts of the ratio by \(4\) (the denominator of the fractions):
- \(3 \times 4 = 12\)
- \(\frac{5}{4} \times 4 = 5\)
- \(\frac{13}{4} \times 4 = 13\)

Thus, the sides of the triangle are \(12 : 5 : 13\).

### Step 4: Check for a Pythagorean Triplet
Next, we check if these sides form a Pythagorean triplet, which is a condition for a right-angled triangle. For sides \(a\), \(b\), and \(c\) (where \(c\) is the longest side), the condition is:
\[ c^2 = a^2 + b^2 \]

Here, let \(a = 5\), \(b = 12\), and \(c = 13\).

Calculating:
- \(c^2 = 13^2 = 169\)
- \(a^2 + b^2 = 5^2 + 12^2 = 25 + 144 = 169\)

Since \(c^2 = a^2 + b^2\), the triangle satisfies the Pythagorean theorem.

### Conclusion
Since the sides \(12\), \(5\), and \(13\) satisfy the Pythagorean theorem, the triangle is a **right-angled triangle**.

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