Learn about altitudes and medians of a triangle through these examples.
Example 1: What is an Altitude?
Define an altitude of a triangle and its properties.
Definition: An **altitude** of a triangle is a perpendicular segment from a vertex to the opposite side (or the line containing the opposite side).
Properties:
Every triangle has three altitudes, one from each vertex.
An altitude forms a right angle (90 degrees) with the side it intersects.
The point where the three altitudes of a triangle intersect is called the **orthocenter (O)**.
Diagram (Acute Triangle):
A
/ \
/ \
/ .O \ (Orthocenter inside)
/ | \
/_____|___\
B D C
AD is an altitude from vertex A to side BC. at D.
Note: The orthocenter's position depends on the type of triangle:
In an acute triangle, the orthocenter is inside the triangle.
In a right triangle, the orthocenter is at the vertex with the right angle.
In an obtuse triangle, the orthocenter is outside the triangle.
Example 2: What is a Median?
Define a median of a triangle and its properties.
Definition: A **median** of a triangle is a segment from a vertex to the midpoint of the opposite side.
Properties:
Every triangle has three medians, one from each vertex.
A median bisects the opposite side (divides it into two equal parts).
The point where the three medians of a triangle intersect is called the **centroid (G)**.
Diagram:
A
/ \
/ \
/ .G \ (Centroid always inside)
/ \
/_____M___\
B C
AM is a median from vertex A to the midpoint M of side BC.
BM = MC.
Note: The centroid is always located inside the triangle. It is also the triangle's center of mass.
Example 3: Drawing an Altitude
Describe how to draw an altitude from a vertex to the opposite side.
Step 1: Choose a vertex (e.g., vertex A) and the opposite side (e.g., side BC).
Step 2: Place a ruler or straightedge along the opposite side (BC).
Step 3: Use a set square (or protractor) to draw a line segment from the vertex (A) that is perpendicular to the side (BC). The set square's right angle should align with the side.
Step 4: The point where the perpendicular segment meets the side (or its extension) is the foot of the altitude. Label this point (e.g., D).
Step 5: The segment from the vertex to the foot of the altitude (AD) is the altitude. Mark the right angle symbol at the intersection.
Note: For obtuse triangles, the foot of the altitude may lie outside the triangle, on the extension of the opposite side.
Example 4: Drawing a Median
Describe how to draw a median from a vertex to the opposite side.
Step 1: Choose a vertex (e.g., vertex A) and the opposite side (e.g., side BC).
Step 2: Find the midpoint of the opposite side (BC). You can do this by measuring the length of the side and dividing by 2, or by using a compass to construct the perpendicular bisector (the midpoint is on the perpendicular bisector).
Step 3: Mark the midpoint of the side (e.g., label it M).
Step 4: Draw a line segment connecting the vertex (A) to the midpoint (M) of the opposite side.
Step 5: The segment AM is the median. You can mark the two equal segments on the opposite side (BM and MC) to indicate the midpoint.
Example 5: Orthocenter vs. Centroid
Summarize the difference between the orthocenter and the centroid.
Orthocenter (O): The point of concurrency (intersection) of the **altitudes** of a triangle.
Centroid (G): The point of concurrency (intersection) of the **medians** of a triangle.
Key Difference: Altitudes are perpendicular to the opposite side, while medians connect to the midpoint of the opposite side.
Location: The orthocenter can be inside, outside, or on the triangle. The centroid is always inside the triangle.
Practice Mode
Ask a question about altitudes, medians, orthocenter, or centroid.
Example Prompts: "What is the orthocenter?", "How is an altitude different from a median?", "Where is the centroid located?".
Related Concepts
Explore these related mathematical concepts to deepen your understanding of triangle properties.
Triangle
A polygon with three sides and three vertices.
Vertex (Vertices)
A corner point of a triangle.
Side
A line segment forming the boundary of a triangle.
Angle
Formed by two sides meeting at a vertex.
Perpendicular
Lines or segments that intersect at a 90-degree angle.
Midpoint
The point exactly halfway along a line segment.
Point of Concurrency
The single point where three or more lines or segments intersect.
Types of Triangles
Classified by sides (scalene, isosceles, equilateral) or angles (acute, right, obtuse).