Question

Landscaping Two triangular gardens have the same size and shape. The landscaper needed 24 feet of fencing for one garden. How much fencing is needed for the second garden? Explain your reasoning.

Short answer

The second garden needs 24 feet of fencing, same as the first garden.

Step-by-step solution

Understanding the Problem

We have two triangular gardens that are congruent, meaning they have the same size and shape. We know that 24 feet of fencing was used for one garden.

Identifying Congruence

Since the two gardens have the same size and shape, they are congruent. This means that all corresponding sides and angles are equal in measurement.

Applying Congruence to Perimeter

The perimeter of a congruent shape is identical to another congruent shape. Since the first garden requires 24 feet of fencing, the second garden, being congruent, requires the same.

Conclusion

The amount of fencing needed for the second garden is the same as for the first garden, due to their congruency.

Key concepts

Understanding Perimeter

The perimeter of a shape is the total distance around the boundary of the shape. For a triangle, the perimeter is the sum of the lengths of all its sides. Calculating the perimeter helps you know how much material, like fencing, you'll need to enclose a space.

• For a triangular garden, if each side measures 'a', 'b', and 'c' feet, the perimeter is given by the formula: \( P = a + b + c \)
• If the perimeter is a specific number, such as 24 feet, it implies that all three sides of the triangle add up to 24 feet.

Understanding this concept is crucial when planning any project that requires enclosing or outlining a space, like a garden.
It helps ensure you have the right amount of resources.

Triangular Gardens Basics

Triangular gardens are a popular landscaping choice due to their aesthetic appeal and space efficiency. These gardens are shaped like a triangle, meaning they have three sides and three angles.

• Each angle in a triangle can be of different measure, but they always add up to 180 degrees.
• For landscaping, knowing the exact shape and sides' measures helps in planning pathways, planting layouts, and fencing.

When dealing with triangular gardens, the congruence of two triangles means that both have identical angles and side lengths. This makes for easy replication of garden designs across similar plots of land.

Congruence in Geometry

Congruence is a concept in geometry where two shapes are identical in form and size. Two congruent triangles have exactly the same three side lengths and angles, although their orientation or placement in space might differ.

• If two triangles are congruent, all their corresponding sides and angles are equal.
• Congruence can be proven by criteria such as Side-Side-Side (SSS), Angle-Side-Angle (ASA), or Side-Angle-Side (SAS).

In practical applications like landscaping, knowing that two gardens are congruent eliminates the need for re-measuring or recalculating for similar designs.
This is why understanding the congruence concept can save time and resources, making project implementation smoother.