Triangle ABC: Calculate the Perpendicular Height from Vertex A

Q: What's the difference between a side and a height of a triangle?

A side connects two vertices of the triangle (like AB, BC, AC), while a height is a perpendicular line from one vertex straight down to the opposite side. Heights always form right angles!

Q: How can I tell which line segment is the height?

Look for the right angle symbol (small square) where the line meets the base. In this diagram, you can see right angle markers where AD meets BC, showing that AD is perpendicular.

Q: Can a triangle have more than one height?

Yes! Every triangle has three heights - one from each vertex to the opposite side. In this problem, we're specifically looking for the height from vertex A.

Q: Why isn't AB or AC the height here?

AB and AC are slanted sides of the triangle. A height must be perpendicular (form a 90° angle) to the base. Only AD drops straight down and forms a right angle with base BC.

Q: What if the height falls outside the triangle?

Sometimes in obtuse triangles, the height can fall outside the triangle when extended. But in this case, the height AD falls inside triangle ABC, making it easy to identify.

Triangle Heights with Perpendicular Identification

Given the following triangle:

Write down the height of the triangle ABC.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Determine the height of the triangle

00:03 The height in a triangle is a perpendicular line from a vertex to its opposite side

00:11 At the intersection point, the angle between the lines is a right angle

00:14 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Given the following triangle:

Write down the height of the triangle ABC.

Step-by-step solution

In the given diagram, we need to determine the height of triangle $\triangle ABC$ . The height of a triangle is defined as the perpendicular segment from a vertex to the line containing the opposite side.

Upon examining the diagram:

Point $A$ is at the top of the triangle.
The side $BC$ is horizontal, lying at the base.
Line segment $AD$ is drawn from point $A$ perpendicularly down to the base $BC$ at point $D$ . This forms a right angle at $D$ with line $BC$ .

Therefore, line segment $AD$ is the perpendicular or the height of triangle $\triangle ABC$ .

Consequently, the height of triangle $\triangle ABC$ is represented by the segment $AD$ .

Final Answer

Key Points to Remember

Essential concepts to master this topic

Definition: Height is the perpendicular segment from vertex to opposite side
Technique: Look for right angle symbol where height meets base
Check: Height must form 90° angle with the base line ✓

Common Mistakes

Avoid these frequent errors

Confusing height with side length
Don't choose AB, AC, or BC as the height = these are sides, not heights! These are slanted lines that don't form right angles with the base. Always choose the perpendicular segment that drops straight down from a vertex to the opposite side.

Practice Quiz

Test your knowledge with interactive questions

Is the straight line in the figure the height of the triangle?

FAQ

Everything you need to know about this question

What's the difference between a side and a height of a triangle?

A side connects two vertices of the triangle (like AB, BC, AC), while a height is a perpendicular line from one vertex straight down to the opposite side. Heights always form right angles!

How can I tell which line segment is the height?

Look for the right angle symbol (small square) where the line meets the base. In this diagram, you can see right angle markers where AD meets BC, showing that AD is perpendicular.

Can a triangle have more than one height?

Yes! Every triangle has three heights - one from each vertex to the opposite side. In this problem, we're specifically looking for the height from vertex A.

Why isn't AB or AC the height here?

AB and AC are slanted sides of the triangle. A height must be perpendicular (form a 90° angle) to the base. Only AD drops straight down and forms a right angle with base BC.

What if the height falls outside the triangle?

Sometimes in obtuse triangles, the height can fall outside the triangle when extended. But in this case, the height AD falls inside triangle ABC, making it easy to identify.

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