In a park, there is right angled triangular flower bed . It's two small sides are 5m &12m respectively . Along its all sides at a distance of 1/2 m each , plants of different types are to be planted . Rose plants are to be planted along the shortest side,Marigold plants are to be planted along the longest side & sunflower plant along the sunflower plant along the third side. At each of the vertex a different type of flower plant is to be planted . A plant is chosen at random . Find the probability that the chosen plant is
(i) On the longest side .
(ii) Sun flower plants .

Answer

To solve the problem step by step, we will first determine the total number of plants planted along each side of the triangular flower bed and then calculate the required probabilities.

### Step 1: Calculate the length of the sides of the triangle
We know the two smaller sides of the right-angled triangle are 5 m and 12 m. We can find the length of the hypotenuse using the Pythagorean theorem.

\[ \text{Hypotenuse} = \sqrt{(5^2 + 12^2)} = \sqrt{(25 + 144)} = \sqrt{169} = 13 \text{ m} \]

### Step 2: Determine the number of plants along each side
Plants are to be planted every 0.5 m along each side.

- **Shortest side (5 m)**:
\[ \text{Number of plants} = \frac{5}{0.5} + 1 = 10 + 1 = 11 \text{ plants} \]

- **Longest side (13 m)**:
\[ \text{Number of plants} = \frac{13}{0.5} + 1 = 26 + 1 = 27 \text{ plants} \]

- **Third side (12 m)**:
\[ \text{Number of plants} = \frac{12}{0.5} + 1 = 24 + 1 = 25 \text{ plants} \]

### Step 3: Calculate the total number of plants
Now, we add the number of plants from all three sides:

\[ \text{Total number of plants} = 11 + 27 + 25 = 63 \text{ plants} \]

### Step 4: Calculate the probability of choosing a plant on the longest side
The probability of choosing a plant on the longest side (Marigold plants) is given by the formula:

\[ \text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{27}{63} \]

This can be simplified:

\[ \frac{27}{63} = \frac{3}{7} \]

### Step 5: Calculate the probability of choosing a sunflower plant
The probability of choosing a sunflower plant (planted along the third side) is:

\[ \text{Probability} = \frac{25}{63} \]

### Final Answers
(i) The probability that the chosen plant is on the longest side is \( \frac{3}{7} \).

(ii) The probability that the chosen plant is a sunflower plant is \( \frac{25}{63} \).

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