Question

Triangle \(D E F\) has sides that measure 6 feet, 6 feet, and 9 feet. Classify the triangle by its sides.

Short answer

The triangle is isosceles.

Step-by-step solution

Understand the Triangle Classification by Sides

Triangles can be classified into three types based on the lengths of their sides: equilateral (all sides equal), isosceles (two sides equal), and scalene (no sides equal). First, identify which category each side belongs to.

Identify the Side Lengths

The given triangle has sides measuring 6 feet, 6 feet, and 9 feet. List them as follows: \( DE = 6 \text{ feet}, EF = 6 \text{ feet}, DF = 9 \text{ feet} \).

Compare the Side Lengths

Check if any sides are equal. Here, the sides \( DE = 6 \text{ feet} \) and \( EF = 6 \text{ feet} \) are equal, while \( DF = 9 \text{ feet} \) is not equal to the other two sides.

Classify the Triangle

Since two sides of the triangle are equal ( DE = EF overDE = 6 and EF = 6 ), the triangle is classified as isosceles.

Key concepts

Triangle Classification

Triangles are fascinating shapes, and classifying them is an essential part of geometry. When we classify triangles, we often look at their sides, angles, or both. However, for the purpose of this discussion, we'll focus on classification by side lengths. There are three specific categories for triangles based on their sides:

• Equilateral Triangle: All three sides have the same length. This type of triangle has three equal angles as well, making it symmetric and balanced.
• Isosceles Triangle: Two sides are of equal length. This unique trait means two of its angles are also equal.
• Scalene Triangle: All sides—and, consequently, all angles—are different. This triangle has no sides of equal length, making it the most asymmetrical of the three.

Understanding the classification helps in simplifying complex problems and provides insights into the properties of these triangles.

Triangle Properties

Triangles, regardless of their shapes, share several common properties that are fundamental in geometry:

• Angle Sum: The sum of the internal angles in every triangle is always 180 degrees. This property is vital in solving for unknown angles when analyzing different problems.
• Base Angles in Isosceles Triangles: In an isosceles triangle, the angles opposite the equal sides—known as the base angles—are also equal. This is one clue used when solving for unknown angles or classifying triangles.
• Exterior Angle Theorem: The measure of an exterior angle of a triangle equals the sum of the two non-adjacent interior angles. This property is beneficial in proofs and problem solving.

Knowing these properties makes it easier to work through triangle-related exercises and apply the correct methods for finding missing measurements, whether they are angles or sides.

Side Lengths of Triangles

The angle and side relationships in triangles are intricate, and understanding them allows for accurate problem-solving. Examining the side lengths is crucial, especially in identifying triangle types:

• Isosceles Triangle Side Lengths: As seen in triangle DEF, having two sides of equal length (6 feet, 6 feet) and one side different (9 feet) is a clear indicator of an isosceles triangle. It's crucial to compare each side to make this determination.
• Triangle Inequality Theorem: This theorem states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side. This serves as a check to ensure the side lengths given can actually form a triangle.
• Pythagorean Theorem for Right Triangles: Although not directly applicable to the isosceles triangle in our exercise, this theorem—for right triangles—equates the square of the hypotenuse to the sum of the squares of the other two sides. It helps categorize triangles based on right angles but not needed for classifying by side alone.

Detailed understanding of these concepts eases the classification and further analysis of triangles in geometric exercises.