Triangle Height Verification: Analyzing the Altitude's Perpendicularity

Q: What makes a line the altitude of a triangle?

An altitude must be a line segment from a vertex that is perpendicular to the opposite side (or its extension). Without the 90° angle, it's just a regular line segment.

Q: Can a triangle have more than one altitude?

Yes! Every triangle has exactly three altitudes - one from each vertex to the opposite side. All three altitudes intersect at a point called the orthocenter.

Q: What if the altitude falls outside the triangle?

This happens in obtuse triangles! The altitude from the obtuse angle vertex will be perpendicular to the extension of the opposite side, not the side itself.

Q: Does the altitude always divide the base into equal parts?

No! The altitude creates a 90° angle but doesn't necessarily bisect the base. Only in isosceles triangles does the altitude from the vertex angle also bisect the base.

Triangle Altitudes with Perpendicularity Requirements

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

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Step-by-step video solution

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00:03 Let's find out if the line in the picture is the height of the triangle.

00:08 A triangle's height starts at one corner and meets the opposite side at a right angle.

00:14 Since this line isn't at a right angle, it's not the height.

00:18 And that's how you solve this problem!

Step-by-step written solution

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Understand the problem

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

Step-by-step solution

To determine if the straight line in the figure is the height of the triangle, we must verify whether it is perpendicular to the base of that triangle.

The height (or altitude) of a triangle is defined as a line segment from a vertex perpendicular to the line containing the opposite side (often referred to as the base).

Upon examining the figure, we see a triangle and a straight line drawn from one vertex towards the opposite side. However, there is no indication or mark suggesting that this line is perpendicular to the base.

Without explicit evidence of perpendicularity, such as a right-angle marking, we cannot assume that the line is the height of the triangle.

Thus, based on the geometric principles related to altitudes in triangles, we conclude the solution to the problem:

No, the straight line in the figure is not the height of the triangle.

Final Answer

Key Points to Remember

Essential concepts to master this topic

Definition: Triangle altitude must be perpendicular to the opposite side
Technique: Look for right angle symbol (⊥) or 90° marking
Check: Without perpendicular marking, line cannot be confirmed as altitude ✓

Common Mistakes

Avoid these frequent errors

Assuming any line from vertex to opposite side is an altitude
Don't assume every line from a vertex to the opposite side is the height = wrong identification! A line must be perpendicular to qualify as an altitude. Always verify perpendicularity with right angle markings or explicit measurements.

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 makes a line the altitude of a triangle?

An altitude must be a line segment from a vertex that is perpendicular to the opposite side (or its extension). Without the 90° angle, it's just a regular line segment.

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

Look for these clues:

A small square symbol (⊥) at the intersection
The marking "90°" near the angle
Explicit statement that the line is perpendicular

Can a triangle have more than one altitude?

Yes! Every triangle has exactly three altitudes - one from each vertex to the opposite side. All three altitudes intersect at a point called the orthocenter.

What if the altitude falls outside the triangle?

This happens in obtuse triangles! The altitude from the obtuse angle vertex will be perpendicular to the extension of the opposite side, not the side itself.

Does the altitude always divide the base into equal parts?

No! The altitude creates a 90° angle but doesn't necessarily bisect the base. Only in isosceles triangles does the altitude from the vertex angle also bisect the base.

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Triangle Height Verification: Analyzing the Altitude's Perpendicularity

Triangle Altitudes with Perpendicularity Requirements

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

What makes a line the altitude of a triangle?

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

Can a triangle have more than one altitude?

What if the altitude falls outside the triangle?

Does the altitude always divide the base into equal parts?

🌟 Unlock Your Math Potential

More Questions

Parts of a Triangle

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Triangle Height Verification: Analyzing a Perpendicular Line in Geometry