Matt has a garden in the shape of a right-angled triangle with one of the acute angles as `30^(@)` . If the longest side of the triangle is 4 m long, what is the perimeter, in meters , of the garden?

Answer

To find the perimeter of Matt's garden, which is in the shape of a right-angled triangle with one acute angle measuring \(30^\circ\) and the longest side (hypotenuse) measuring 4 meters, we can follow these steps:

### Step 1: Identify the triangle's angles and sides
In a right-angled triangle, if one of the acute angles is \(30^\circ\), the other acute angle must be \(60^\circ\) (since \(90^\circ + 30^\circ + 60^\circ = 180^\circ\)). The sides of a \(30^\circ\)-\(60^\circ\)-\(90^\circ\) triangle have a specific ratio:
- The side opposite the \(30^\circ\) angle is \(x\),
- The side opposite the \(60^\circ\) angle is \(x\sqrt{3}\),
- The hypotenuse (opposite the \(90^\circ\) angle) is \(2x\).

### Step 2: Set up the equation for the hypotenuse
Given that the hypotenuse is 4 meters, we can set up the equation:
\[ 2x = 4 \]

### Step 3: Solve for \(x\)
To find \(x\), divide both sides of the equation by 2:
\[ x = \frac{4}{2} = 2 \]

### Step 4: Calculate the lengths of the other two sides
Now that we have \(x = 2\):
- The side opposite the \(30^\circ\) angle (which is \(x\)) is:
\[ 2 \text{ meters} \]
- The side opposite the \(60^\circ\) angle (which is \(x\sqrt{3}\)) is:
\[ 2\sqrt{3} \text{ meters} \]

### Step 5: Calculate the perimeter
The perimeter \(P\) of the triangle is the sum of all three sides:
\[ P = \text{(side opposite } 30^\circ) + \text{(side opposite } 60^\circ) + \text{(hypotenuse)} \]
\[ P = 2 + 2\sqrt{3} + 4 \]
\[ P = 6 + 2\sqrt{3} \]

### Step 6: Approximate the value of the perimeter
To get a numerical approximation, we can calculate:
\[ 2\sqrt{3} \approx 2 \times 1.732 \approx 3.464 \]
Thus,
\[ P \approx 6 + 3.464 = 9.464 \text{ meters} \]

### Final Answer
The perimeter of the garden is approximately \(9.46\) meters.

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