Identifying Triangle Height: Geometric Line Analysis Problem

Q: How can I tell if a line is perpendicular in a diagram?

Look for a small square symbol at the intersection point! This indicates a 90° angle. If there's no square symbol or perpendicular marking, you cannot assume the line is a height.

Q: Can a triangle have more than one height?

Yes! Every triangle has exactly three heights - one from each vertex perpendicular to its opposite side. Some may fall outside the triangle in obtuse triangles.

Q: What's the difference between height and median?

A height goes from vertex perpendicular to the opposite side. A median goes from vertex to the midpoint of the opposite side. They're usually different lines!

Q: Does the height always stay inside the triangle?

Not always! In obtuse triangles, two of the heights fall outside the triangle. They still go from the vertex perpendicular to the extended opposite side.

Q: Why is perpendicularity so important for heights?

Heights are used to calculate triangle area using the formula: $ Area = \frac{1}{2} \times base \times height $. Only perpendicular lines give the true shortest distance needed for accurate area calculation.

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Determine whether the line is a height in the triangle

00:04 A height in a triangle creates a right angle with the line that it intersects

00:10 Draw the line that creates a right angle with the side

00:19 We can observe that our angle is smaller than a right angle

00:23 Therefore, the line is not a height

00:28 These lines are heights (they create right angles)

00:37 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

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

2

Step-by-step solution

To solve this problem, we must determine whether the dashed line in the presented triangle fulfills the criteria of being a height. Let's verify each critical aspect:

First, identify what a height (altitude) is: It is a line drawn from a vertex perpendicular to the opposite side.
Next, observe the figure. We have a triangle with a dash line drawn from the top vertex to a side that seems to extend from one corner of the base to another point on the extended base.
Since this line is not shown to be perpendicular to the base of the triangle (no right angle box), we cannot affirm that it fulfills our requirement.

As a result, the straight line does not meet the standard definition of a height for this triangle since it does not form the necessary 90-degree angle with the base. Therefore, as the line is not perpendicular to the opposite side, it is not the height.

Thus, the correct answer to the problem is No.

3

Final Answer

No

Key Points to Remember

Essential concepts to master this topic

Definition: Height is a line from vertex perpendicular to opposite side
Technique: Look for right angle symbol (small square) at intersection point
Check: Verify the line forms 90° angle with the base ✓

Common Mistakes

Avoid these frequent errors

Assuming any line from vertex to opposite side is a height
Don't think every line from a vertex to the opposite side is automatically a height = wrong identification! A height must be perpendicular, forming exactly 90°. Always check for the right angle symbol or perpendicular indication.

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

How can I tell if a line is perpendicular in a diagram?

+

Look for a small square symbol at the intersection point! This indicates a 90° angle. If there's no square symbol or perpendicular marking, you cannot assume the line is a height.

Can a triangle have more than one height?

+

Yes! Every triangle has exactly three heights - one from each vertex perpendicular to its opposite side. Some may fall outside the triangle in obtuse triangles.

What's the difference between height and median?

+

A height goes from vertex perpendicular to the opposite side. A median goes from vertex to the midpoint of the opposite side. They're usually different lines!

Does the height always stay inside the triangle?

+

Not always! In obtuse triangles, two of the heights fall outside the triangle. They still go from the vertex perpendicular to the extended opposite side.

Why is perpendicularity so important for heights?

+

Heights are used to calculate triangle area using the formula: $Area = \frac{1}{2} \times base \times height$ . Only perpendicular lines give the true shortest distance needed for accurate area calculation.

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Identifying Triangle Height: Geometric Line Analysis Problem

Triangle Heights with Perpendicularity Requirements

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