Question

The lengths of three sides of a triangle are given. Determine whether each triangle is a right triangle. \(1 \mathrm{~mm}, 1 \mathrm{~mm}, \sqrt{2} \mathrm{~mm}\)

Short answer

Yes, the triangle is a right triangle.

Step-by-step solution

Identify the sides

The sides of the triangle are given as 1 mm, 1 mm, and \( \sqrt{2} \) mm. These sides represent a potential right triangle, where \( \sqrt{2} \) mm could be the hypotenuse because it is the largest of the three.

Apply the Pythagorean theorem

To determine if the triangle is a right triangle, use the Pythagorean theorem: \( a^2 + b^2 = c^2 \), where \( c \) is the hypotenuse. Here, substitute the side lengths: \( 1^2 + 1^2 = (\sqrt{2})^2 \).

Calculate the squares

Calculate the squares of each side: \( 1^2 = 1 \), so \( 1 + 1 = 2 \). Also, \( (\sqrt{2})^2 = 2 \).

Compare the results

Compare the sum of the squares of the two shorter sides with the square of the hypotenuse: \( 2 = 2 \). Since they are equal, this confirms that the triangle satisfies the Pythagorean theorem.

Key concepts

Pythagorean Theorem

The Pythagorean theorem is a remarkable principle that is central to many geometric problems. It's named after the ancient Greek mathematician Pythagoras and it's used to find out if a triangle is a right triangle.
This theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. The formula is expressed as: \[ a^2 + b^2 = c^2 \]where \( c \) is the hypotenuse, and \( a \) and \( b \) are the lengths of the other two sides.
This relationship holds true only for right triangles, which makes it a handy tool for verifying if a triangle is right-angled based on the lengths of its sides. Recognizing and applying this theorem is fundamental to solving many geometry problems.

Triangle Sides

Each triangle is formed by three sides, and identifying these sides correctly is important for many calculations. In the specific case of right triangles, notably:

• The hypotenuse is always the longest side.
• The other two sides are referred to as the 'legs' of the triangle.

This helps in applying the Pythagorean theorem accurately. For any question concerning whether a given set of side lengths forms a right triangle, always ensure that the largest number is considered as the potential hypotenuse. In problems like the original exercise, recognizing \( \sqrt{2} \) mm as the longest side is crucial in confirming that this set of sides makes up a right triangle. Correctly identifying the relationships between the sides is important in further geometric calculations and problem-solving.

Geometry Problem Solving

Solving geometry problems, particularly when involving triangles, often relies on a systematic approach to verifying triangle relationships. Here's how:

• Begin by thoroughly understanding the given information, such as the lengths of the sides.
• Identify which side could potentially be the hypotenuse based on length.
• Apply relevant formulas such as the Pythagorean theorem to test your hypothesis.
• Check calculations step-by-step to confirm the outcome coincides with theoretical predictions.

When these steps are applied carefully, even complex problems can be handled efficiently. In the section above, using the Pythagorean theorem unequivocally showed that the provided set of sides does indeed form a right triangle. By adopting a structured approach to problem-solving, you can be confident in handling different variations of geometric problems.