Problem 57 - Precalculus

Question

If you are given two sides that have the same length in a triangle, then there can be at most one triangle.

Step-by-Step Solution

Answer

In a triangle with two equal sides, only one triangle can exist for a given third side.

Step 1: Understand the Given Information

In a triangle, if two sides are given to be of the same length, it implies we have an isosceles triangle. This means two sides are equal and we need to determine if there can be multiple triangles formed with the same given conditions.

Step 2: Recall Properties of Isosceles Triangle

Recall that in an isosceles triangle, the two angles opposite the equal sides are also equal. The properties of a triangle also state that the sum of the angles in any triangle must be 180 degrees.

Step 3: Establish Relationship and Constraints

Name the sides as follows: let the equal sides be 'a', and the base, the third side, be 'b'. In an isosceles triangle, the angles opposite to 'a' are equal. Denote these angles as \( \theta \). The remaining angle will be \( 180^ - 2\theta \). Compute all using the properties and Pythagorean Theorem if necessary.

Step 4: Determine Uniqueness of Triangle

The key factor making the triangle unique is the fixed angles and side lengths. Given two sides 'a' are equal in length, the possible angles are determined solely by the length of the third side 'b'. Hence, fixing the length of side 'b' would mean there is only one triangle possible for the given set of sides 'a', 'a', and 'b'.

Key Concepts

Triangle Properties

Triangles are one of the fundamental shapes in geometry. When studying triangles, several key properties must be understood:

• Triangles are closed shapes with three sides, three vertices, and three angles.
• The sum of the internal angles in any triangle is always 180 degrees.
• The triangle inequality principle states that any side of a triangle must be less than the sum of the other two sides for the shape to be a valid triangle.

In the context of an isosceles triangle, two of its sides are of equal length. This specific characteristic leads to additional properties, such as multiple equal angles. Each type of triangle, whether it is scalene, isosceles, or equilateral, follows these basic rules, but with its unique twists due to its side-length configuration.

Triangle Uniqueness

Every triangle is unique based on the length of its sides and the measure of its angles. For isosceles triangles, uniqueness emerges starkly due to the conditions outlined by its properties:

• If two sides are given to be of equal length, the triangle's base and the angles it forms are the deciding factors for its uniqueness.
• An isosceles triangle can have only one possible form as the set lengths of its sides dictate exact angle measures.
• Fixing the base length in an isosceles triangle means that no other distinct triangle can be created with the same dimensions without altering a fundamental property.

Hence, when given two equal sides and a specific base, only one unique isosceles triangle can be formed.

Isosceles Triangle Angles

The angles in an isosceles triangle are closely tied to its side lengths. This characteristic leads to some intriguing outcomes:

• In any isosceles triangle, the angles opposite the equal sides are identical. Let's denote these angles as \( \theta \).
• Given that the sum of angles in any triangle is 180 degrees, we can compute: \( 2\theta + \text{base angle} = 180^\circ \).
• The base angle can be found by subtracting the sum of the equal angles from 180 degrees, represented as: \( 180^\circ - 2\theta \).

With these calculations, specific angles are defined solely by the length of the triangle's sides. This means in an isosceles configuration, once side lengths are established, the angles follow, ensuring the triangle's form is consistent and reliable.