Lucas is designing a flower garden in the shape of an isosceles right triangle. He has created a scale diagram. The lengths of the perpendicular sides in the scale diagram are 7 cm, and the hypotenuse of the real garden will be 3 m long. What is the area of the real garden?

answered

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To find the area of the real garden, we need to determine the dimensions of the garden using the scale diagram provided and then use those dimensions to calculate the area.

Step 1: Determine the scale factor
The scale factor is the ratio between the lengths on the scale diagram and the corresponding lengths in reality. Here, the length of the perpendicular side on the scale diagram is 7 cm, while the hypotenuse of the real garden will be 3 m (which is equivalent to 300 cm). Therefore, the scale factor is 300 cm / 7 cm.

Step 2: Calculate the lengths of the perpendicular sides
To determine the lengths of the perpendicular sides, we will multiply the lengths from the scale diagram by the scale factor.

Perpendicular side 1: 7 cm × (300 cm / 7 cm) = 300 cm
Perpendicular side 2: 7 cm × (300 cm / 7 cm) = 300 cm

Step 3: Calculate the area of the real garden
The area of a right triangle can be calculated using the formula: Area = (base × height) / 2. In this case, we can use either of the perpendicular sides as the base and the other perpendicular side as the height.

Using Perpendicular side 1 as the base and Perpendicular side 2 as the height:
Area = (300 cm × 300 cm) / 2 = 45,000 cm²

Alternatively, we can use Perpendicular side 2 as the base and Perpendicular side 1 as the height, and we will get the same result.

Therefore, the area of the real garden is 45,000 cm².