The sides of a triangular park are in the ratio 3 : 4 : 5 and the perimeter of the park is 170 metres. What is the length, in metres, of the perpendicular drawn from the vertex opposite the largest side to that largest side?

Introduction / Context:
This geometry question involves a triangle whose sides are in the ratio 3 : 4 : 5 and whose perimeter is known. The ratio 3 : 4 : 5 is characteristic of a right angled triangle, where the side with length 5 parts is the hypotenuse and the other two sides are the perpendicular legs. We are asked to find the perpendicular distance from the vertex opposite the largest side (the hypotenuse) to that side. In a right triangle, this distance is simply the altitude to the hypotenuse, which can be found using area relationships.

Given Data / Assumptions:

• Side ratios of the triangle are 3 : 4 : 5.
• Perimeter of the triangle = 170 metres.
• The largest side is the one corresponding to 5 parts.
• We need the perpendicular distance from the opposite vertex to this largest side.
• The triangle is assumed to be a 3 : 4 : 5 right triangle.

Concept / Approach:
If the sides of a triangle are in the ratio 3 : 4 : 5, it represents a right triangle with the largest side as the hypotenuse. Let the common multiplying factor be k. Then the sides are 3k, 4k and 5k. The perimeter then is 12k. From the given perimeter we can find k and therefore the actual side lengths. The area of a right triangle is (1 / 2) * (product of the two legs). The altitude from the right angle to the hypotenuse can then be found using the formula: altitude = (2 * area) / hypotenuse.

Step-by-Step Solution:
Let the sides be 3k, 4k and 5k metres. Perimeter = 3k + 4k + 5k = 12k. Given perimeter = 170 metres, so 12k = 170. Thus k = 170 / 12 = 85 / 6. Then the three sides are: 3k = 3 * 85 / 6 = 255 / 6 metres, 4k = 340 / 6 metres and 5k = 425 / 6 metres. The triangle is right angled with legs 3k and 4k and hypotenuse 5k. Area of triangle = (1 / 2) * (3k) * (4k) = 6k^2. Altitude from the right angle to the hypotenuse (largest side) = (2 * area) / hypotenuse. So altitude = (2 * 6k^2) / (5k) = (12k^2) / (5k) = 12k / 5. Substitute k = 85 / 6: altitude = 12 * (85 / 6) / 5. Simplify: 12 * (85 / 6) = 2 * 85 = 170, so altitude = 170 / 5 = 34 metres.

Verification / Alternative check:
We can approximate the side lengths numerically: 3k ≈ 42.5, 4k ≈ 56.67 and 5k ≈ 70.83 metres. Using the formula for area from base and height, area = (1 / 2) * hypotenuse * altitude. With altitude 34 and hypotenuse about 70.83, area ≈ (1 / 2) * 70.83 * 34 ≈ 1202. The area from legs is (1 / 2) * 42.5 * 56.67 ≈ 1202 as well, within rounding. This confirms that altitude 34 metres is consistent.

Why Other Options Are Wrong:
Values like 24, 30, 36 or 40 metres do not satisfy the relationship area = (1 / 2) * base * height when used with the actual hypotenuse and the legs derived from the given perimeter. Only altitude 34 metres makes the two different area expressions match.

Common Pitfalls:
Some candidates mistake the largest side as one of the legs rather than the hypotenuse, leading to wrong use of the Pythagoras theorem. Others incorrectly use half the perimeter as the hypotenuse or mix up the altitude with a leg length. Another error is to ignore that the ratio 3 : 4 : 5 implies a right triangle. Proper recognition of the 3 : 4 : 5 pattern and careful algebra with k removes confusion.

Final Answer:
The perpendicular distance from the vertex to the largest side is 34 metres.