Introduction

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Understanding the various components and relationships within a right triangle is essential in geometry and trigonometry. One such relationship is provided by Pythagoras' Theorem, which allows us to find the lengths of the sides of a right triangle given the lengths of the other two sides. This theorem is particularly useful when we know the hypotenuse and one other side of the triangle.

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Understanding the Hypotenuse and Legs of a Right Triangle

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A right triangle is a triangle that contains one right angle (90 degrees). The sides opposite these angles are given special names: the hypotenuse is the longest side and is opposite the right angle, while the other two sides are called legs. The theorem named after the ancient Greek mathematician Pythagoras states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs. Mathematically, it can be expressed as:

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Formula:

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c2 a2 b2

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where c represents the length of the hypotenuse, and a and b represent the lengths of the legs.

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Problem Statement

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In the original problem, it was mentioned that the hypotenuse of a right triangle is 150 scales, and one of the legs is 2 cm. This combination of units makes the problem unclear because '150 scales' is not a recognized unit of measurement. To proceed with the calculation, we will need to provide a more consistent measurement for the hypotenuse, most likely in centimeters (cm).

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Assuming that the hypotenuse of 150 scales should be converted to centimeters (considering that 1 scale is approximately 1 cm), the hypotenuse would then be 150 cm. With this in mind, let's calculate the length of the other leg using Pythagoras' Theorem.

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Calculating the Second Leg of the Triangle

r r r Substitute the known values into the Pythagorean Theorem:r r Given: c 150 cm (hypotenuse), a 2 cm (one leg)r c2 a2 b2r 1502 22 b2r 22500 4 b2r b2 22496r b √22496 ≈ 150.0067 cmr r The length of the other leg (b) is approximately 150.0067 cm.r r r

Calculating the Area of the Triangle

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Now that we have the lengths of the two legs of the right triangle, we can calculate its area. The formula for the area of a right triangle is:

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Formula:

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Area 0.5 × a × b

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Substituting the values we have:

r r r Area 0.5 × 2 cm × 150.0067 cmr Area 150.0067 cm2r r r

Therefore, the area of the right triangle is approximately 150.0067 square centimeters.

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Conclusion

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Using Pythagoras' Theorem, we were able to find the length of the other leg of the triangle and subsequently calculate the area of the right triangle. It is important to ensure that all units are consistent when applying such theorems, as any discrepancies can lead to incorrect results.

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