Question

The lengths of three sides of a triangle are given. Determine whether each triangle is a right triangle. . \(11 \mathrm{~cm}, 60 \mathrm{~cm}, 61 \mathrm{~cm}\)

Short answer

The triangle with sides 11 cm, 60 cm, and 61 cm is a right triangle.

Step-by-step solution

Identify the longest side

The given sides are 11 cm, 60 cm, and 61 cm. To check if it is a right triangle, first identify the longest side, as this will be the hypotenuse if the triangle is right-angled. Here, the longest side is 61 cm.

Apply the Pythagorean Theorem

For a triangle to be right-angled, the Pythagorean Theorem must hold: \(a^2 + b^2 = c^2\), where \(c\) is the longest side or hypotenuse. Here, \(a = 11\), \(b = 60\), and \(c = 61\).

Calculate squares of the sides

Calculate \(a^2\), \(b^2\), and \(c^2\). \(11^2 = 121\)\(60^2 = 3600\)\(61^2 = 3721\)

Compare sums of squares

Now add \(a^2\) and \(b^2\): \(121 + 3600 = 3721\).

Conclude about the right triangle

Since \(a^2 + b^2 = c^2\) (i.e., \(3721 = 3721\)), the given triangle is a right triangle.

Key concepts

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry, specifically related to right triangles. It 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. This can be expressed with the formula:\[a^2 + b^2 = c^2\]Here, \(a\) and \(b\) represent the triangle's two shorter sides, and \(c\) is always the hypotenuse. This theorem is crucial for determining whether a triangle is right-angled when the side lengths are known. For example, using side lengths of 11 cm, 60 cm, and 61 cm, if \(11^2 + 60^2 = 61^2\) holds true, the triangle is indeed a right triangle. This theorem not only helps in identifying right triangles but also forms the basis for many applications in trigonometry and geometry.

Hypotenuse

In triangle geometry, the hypotenuse is the longest side of a right triangle. It is always opposite the right angle. Identifying the hypotenuse is crucial when applying the Pythagorean Theorem, as the equation is centered around this side being equal to the sum of the squares of the other two sides.

• The hypotenuse is labeled as \(c\) in the Pythagorean Theorem equation \(a^2 + b^2 = c^2\).
• In a right triangle, the hypotenuse is always opposite the right angle.
• It is essential to correctly identify the hypotenuse to apply the theorem correctly and verify the triangle's dimensions.

For a given set of triangle sides like 11 cm, 60 cm, and 61 cm, the side measuring 61 cm is identified as the hypotenuse. Correctly identifying this side allows the application of the Pythagorean Theorem to determine if the triangle is right-angled.

Triangle Geometry

In geometry, triangles come in various types, but a right triangle holds a special place due to its unique properties. Triangle geometry involves understanding angles and sides to classify them and apply various theorems.

• Right Triangle: One angle equals 90 degrees.
• Isosceles Triangle: Two sides are of equal length.
• Equilateral Triangle: All sides and angles are equal.

A right triangle can be identified using the Pythagorean Theorem, where the presence of a hypotenuse, as the longest side, helps in verifying the right angle. In any given triangle, the sum of angles will always equal 180 degrees. Recognizing these geometric properties is instrumental in identifying and working with various triangles and their specific characteristics, like in the exercise where sides are tested to confirm the right-angle presence.