Question

Draw two types of triangles in which the altitude is on the line that forms the perpendicular bisector. Identify the types of triangles drawn, and draw the altitude and perpendicular bisector for each triangle.

Short answer

Equilateral and isosceles triangles both have altitudes that coincide with their perpendicular bisectors.

Step-by-step solution

Draw an Equilateral Triangle

An equilateral triangle is a triangle where all three sides are of equal length. Start by drawing a triangle with each side of equal length, ensuring that all angles are 60 degrees. This is because the perpendicular bisector and the altitude coincide at one of the triangle's vertices.

Identify Altitude and Perpendicular Bisector in Equilateral Triangle

For an equilateral triangle, the altitude from any vertex to the opposite side also acts as a perpendicular bisector because it cuts the opposite side equally into two halves and creates a right angle. Draw a line from one vertex to the midpoint of the opposite side, ensuring it is perpendicular.

Draw an Isosceles Triangle

An isosceles triangle has two sides of equal length. Draw a triangle such that two sides are equal with a distinct vertex at the top, and the base forms a different length compared to the other two sides. This will ensure that the perpendicular bisector and altitude fall on the same line.

Identify Altitude and Perpendicular Bisector in Isosceles Triangle

In an isosceles triangle, the line from the vertex angle to the midpoint of the base (the line that bisects the base perpendicularly) acts as both the altitude and the perpendicular bisector. Draw this line, ensuring it meets the base at a right angle.

Key concepts

Triangles

Triangles are one of the most basic shapes in geometry. They are polygonal shapes with three sides and three angles. This simplicity makes them both approachable and essential to understand.
Each triangle is classified based on its sides or angles. For instance:

• Equilateral Triangle: All three sides and angles are equal.
• Isosceles Triangle: Two sides and the angles opposite those sides are equal.
• Scalene Triangle: All sides and angles are different.

Knowing how to classify triangles is pivotal when solving geometric problems. This is because each type of triangle has specific properties that can simplify calculations or constructions. A deep understanding of triangles allows learners to make accurate predictions about angles and side lengths when given certain conditions.

Perpendicular Bisector

A perpendicular bisector is a specific type of line that helps in creating symmetry in a triangle. It divides a line segment into two equal parts at a 90-degree angle. Specifically for triangles, it often involves one of the sides being split into two equal sections.
Here are some key points:

• It always forms a right angle with the line segment it bisects.
• In certain triangles like equilateral and some isosceles variants, the perpendicular bisector also acts as the altitude.
• It's useful for proving congruency and symmetry in geometric shapes.

Understanding how to construct and use perpendicular bisectors can help in accurately dividing shapes, and it plays a crucial role in solving many geometric problems such as determining line congruence and distances.

Altitude

The altitude in a triangle is a line segment from a vertex, forming a perpendicular connection to the opposite side, or an extension of that side. Altitude is crucial for understanding triangle height.
Key characteristics include:

• It increases understanding of triangle dimensions, forming right angles with the base.
• In equilateral and some isosceles triangles, altitudes can also be perpendicular bisectors.
• It's essential for calculating triangle area, as area is half the base times the height.

Altitudes help break down triangles into smaller, often right triangles, which simplifies solving complex geometric problems. Learners should be comfortable with locating and drawing altitudes as they provide critical insights into triangle properties.

Equilateral Triangle

An equilateral triangle is particularly symmetrical because all its sides and angles are congruent. This symmetry offers a wealth of geometric properties and relationships that can simplify complex problems.
Important features to keep in mind are:

• All interior angles measure 60 degrees, making them uniformly equal.
• The altitude acts as the perpendicular bisector, dividing the triangle into two congruent right triangles.
• Its regularity makes it a common figure in geometry problems, from basic calculations to architectural design.

By understanding equilateral triangles, learners can navigate other geometric challenges with more confidence. It's crucial to leverage the inherent symmetry of equilateral triangles to identify relationships and congruences quickly.

Isosceles Triangle

In geometry, an isosceles triangle has a distinctive feature where two sides are the same length, creating equal opposite angles. This makes it somewhat symmetrical, yet varied enough from equilateral triangles to offer a different set of useful properties.
Consider the following characteristics:

• Two sides are congruent, making the base angles equal too.
• The altitude from the vertex angle to the base also serves as the perpendicular bisector.
• They provide interesting geometric challenges where known side lengths and angles help determine other triangle dimensions.

Understanding isosceles triangles is key in solving problems with limited information about sides or angles, as their properties can help infer unknown elements. It’s this mix of symmetry and variability that makes them intriguing yet manageable to work with.