What is th area of a right angled traiangle ?
A. The perimeter of the triangle is 30 cm.
B. The ratio between the base and the height of the triangle is `5:12.`
C. The area of the trangle is equal to the area of a reactangle of length 10 cms.

Options

A. Only B and C together are required
B. Only A and B together are required
C. Only either A or B and C together are required.
D. Only A and C together are required

Explanation

To find the area of the right-angled triangle, we will analyze the provided statements step by step.

### Step 1: Understanding the Area of a Right-Angled Triangle
The area \( A \) of a right-angled triangle can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]

### Step 2: Evaluating Statement A
**Statement A:** The perimeter of the triangle is 30 cm.

The perimeter \( P \) of a triangle is given by the sum of all its sides. For a right-angled triangle with base \( b \), height \( h \), and hypotenuse \( c \):
\[ P = b + h + c = 30 \]

However, knowing just the perimeter does not provide us with the base and height directly. We need more information to find the area.

**Conclusion for A:** Insufficient alone.

### Step 3: Evaluating Statement B
**Statement B:** The ratio between the base and the height of the triangle is \( 5:12 \).

Let’s denote the base as \( 5x \) and the height as \( 12x \) for some \( x \).

Now, we can express the area using the base and height:
\[ A = \frac{1}{2} \times (5x) \times (12x) = \frac{1}{2} \times 60x^2 = 30x^2 \]

However, without knowing the value of \( x \), we cannot determine the exact area.

**Conclusion for B:** Insufficient alone.

### Step 4: Evaluating Statement C
**Statement C:** The area of the triangle is equal to the area of a rectangle of length 10 cm.

Let’s denote the breadth of the rectangle as \( b \). The area of the rectangle is:
\[ \text{Area}_{\text{rectangle}} = \text{length} \times \text{breadth} = 10 \times b \]

Since we do not know the breadth \( b \), we cannot determine the area of the triangle from this statement alone.

**Conclusion for C:** Insufficient alone.

### Step 5: Combining Statements A and B
Now, let’s combine Statements A and B.

From Statement B, we have:
- Base = \( 5x \)
- Height = \( 12x \)

From Statement A, we know:
\[ 5x + 12x + c = 30 \]
Where \( c \) can be calculated using the Pythagorean theorem:
\[ c = \sqrt{(5x)^2 + (12x)^2} = \sqrt{25x^2 + 144x^2} = \sqrt{169x^2} = 13x \]

Substituting \( c \) into the perimeter equation:
\[ 5x + 12x + 13x = 30 \]
\[ 30x = 30 \]
\[ x = 1 \]

Now substituting \( x \) back to find base and height:
- Base = \( 5x = 5 \times 1 = 5 \) cm
- Height = \( 12x = 12 \times 1 = 12 \) cm

Now we can calculate the area:
\[ A = \frac{1}{2} \times 5 \times 12 = \frac{60}{2} = 30 \text{ cm}^2 \]

**Conclusion for A and B together:** Sufficient to find the area.

### Step 6: Combining Statements A and C
Let’s check if A and C together provide sufficient information.

From A, we know the perimeter is 30 cm, but we still need the base and height to find the area. Statement C does not help us determine the base or height directly.

**Conclusion for A and C together:** Insufficient.

### Step 7: Combining Statements B and C
Now let’s check if B and C together provide sufficient information.

From B, we have the ratio of base and height, but we still need a specific value to calculate the area. Statement C gives us the area in relation to a rectangle, but without knowing the breadth, we cannot determine the area of the triangle.

**Conclusion for B and C together:** Insufficient.

### Final Conclusion
Only Statements A and B together are sufficient to determine the area of the triangle.

**Answer:** The area of the triangle can be determined using statements A and B together.

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