Triangle Height Analysis: Verifying the Altitude in a Given Triangle

Q: How can I tell if a line is perpendicular to the base?

Look for the right angle symbol (small square) where the line meets the base. This symbol shows the angle is exactly $ 90° $. Without this marker, the line is not an altitude.

Q: Can a triangle have more than one altitude?

Yes! Every triangle has three altitudes - one from each vertex to its opposite side. Each altitude must be perpendicular to the side it meets.

Q: What if the altitude falls outside the triangle?

In obtuse triangles, two altitudes can fall outside the triangle. You extend the base line, and the altitude is still perpendicular to that extended line.

Q: Does the altitude always go to the middle of the base?

No! The altitude goes perpendicular to the base, not necessarily to its midpoint. Only in isosceles triangles does the altitude from the vertex angle bisect the base.

Q: Why is perpendicularity so important for altitudes?

Perpendicularity ensures the shortest distance from the vertex to the opposite side. This is crucial for calculating the triangle's area using $ Area = \frac{1}{2} \times base \times height $.

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:04 Let's decide if this line is a height of the triangle.

00:07 A triangle's height starts at one corner.

00:11 It must also meet the opposite side at a right angle.

00:15 So, this line isn't the height. And that's our 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

In the given problem, we have a triangle depicted with a specific line drawn inside it. The question asks if this line represents the height of the triangle. To resolve this question, we need to discern whether the line is perpendicular to one of the sides of the triangle when extended, as only a line that is perpendicular from a vertex to its opposite side can be considered the height.

The line in question is shown intersecting one of the sides within the triangle but does not form a perpendicular angle with any side shown or the ground (as is required for it to be the height of the triangle). A proper height would typically intersect perpendicularly either at or along the extended line of the opposite side from a vertex.

Therefore, based on the visual clues provided and the typical geometric definition of a height (or altitude) in a triangle, this specific line does not fit the criteria for being a height.

Thus, we conclude that the line depicted is not the height of the triangle. The correct answer is No.

3

Final Answer

No

Key Points to Remember

Essential concepts to master this topic

Definition: Altitude must be perpendicular to the base it meets
Recognition: Look for right angle symbol (square) at base intersection
Verification: Check if line forms 90° angle with opposite side ✓

Common Mistakes

Avoid these frequent errors

Assuming any line from vertex to opposite side is altitude
Don't think every line from a vertex to the opposite side is an altitude = wrong identification! The line must be perpendicular to form a 90° angle. Always check for the perpendicular symbol or right angle marker.

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 to the base?

+

Look for the right angle symbol (small square) where the line meets the base. This symbol shows the angle is exactly $90°$ . Without this marker, the line is not an altitude.

Can a triangle have more than one altitude?

+

Yes! Every triangle has three altitudes - one from each vertex to its opposite side. Each altitude must be perpendicular to the side it meets.

What if the altitude falls outside the triangle?

+

In obtuse triangles, two altitudes can fall outside the triangle. You extend the base line, and the altitude is still perpendicular to that extended line.

Does the altitude always go to the middle of the base?

+

No! The altitude goes perpendicular to the base, not necessarily to its midpoint. Only in isosceles triangles does the altitude from the vertex angle bisect the base.

Why is perpendicularity so important for altitudes?

+

Perpendicularity ensures the shortest distance from the vertex to the opposite side. This is crucial for calculating the triangle's area using $Area = \frac{1}{2} \times base \times height$ .

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Triangle Height Analysis: Verifying the Altitude in a Given Triangle

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