The height and base of a right angle triangle is `15sqrt3` cm and `20sqrt3` cm respectively. Find the perimeter of the triangle.
(a)`45sqrt3cm`
(b)`50sqrt3cm`
(c)`55sqrt3cm`
(d)`60sqrt3cm`

Options

A. `45sqrt3cm`
B. `50sqrt3cm`
C. `55sqrt3cm`
D. `60sqrt3cm`

Answer

To find the perimeter of the right-angled triangle with given height and base, we can follow these steps:

### Step 1: Identify the height and base
The height (perpendicular side) of the triangle is given as \(15\sqrt{3}\) cm, and the base (the other perpendicular side) is given as \(20\sqrt{3}\) cm.

### Step 2: Use the Pythagorean theorem to find the hypotenuse
In a right-angled triangle, the relationship between the sides is given by the Pythagorean theorem:
\[ c^2 = a^2 + b^2 \]
where \(c\) is the hypotenuse, and \(a\) and \(b\) are the other two sides (height and base).

Substituting the values:
\[ c^2 = (15\sqrt{3})^2 + (20\sqrt{3})^2 \]

Calculating each term:
\[ (15\sqrt{3})^2 = 15^2 \cdot 3 = 225 \cdot 3 = 675 \]
\[ (20\sqrt{3})^2 = 20^2 \cdot 3 = 400 \cdot 3 = 1200 \]

Now, adding these values:
\[ c^2 = 675 + 1200 = 1875 \]

### Step 3: Calculate the hypotenuse
Now, we take the square root to find \(c\):
\[ c = \sqrt{1875} \]

To simplify \(\sqrt{1875}\):
\[ \sqrt{1875} = \sqrt{25 \cdot 75} = \sqrt{25} \cdot \sqrt{75} = 5\sqrt{75} \]
Further simplifying \(\sqrt{75}\):
\[ \sqrt{75} = \sqrt{25 \cdot 3} = 5\sqrt{3} \]
Thus,
\[ c = 5 \cdot 5\sqrt{3} = 25\sqrt{3} \text{ cm} \]

### Step 4: Calculate the perimeter
The perimeter \(P\) of the triangle is the sum of all its sides:
\[ P = \text{height} + \text{base} + \text{hypotenuse} \]
Substituting the values:
\[ P = 15\sqrt{3} + 20\sqrt{3} + 25\sqrt{3} \]
Combining like terms:
\[ P = (15 + 20 + 25)\sqrt{3} = 60\sqrt{3} \text{ cm} \]

### Conclusion
Thus, the perimeter of the triangle is \(60\sqrt{3}\) cm.

### Answer
The correct option is (d) \(60\sqrt{3}\) cm.

---