Triangle Classification: When Height Meets Median at Point D

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

A height is perpendicular to the opposite side, while a median connects a vertex to the midpoint of the opposite side. When the same line does both jobs, something special happens!

Q: Why does having both properties make it isosceles?

When AD is both height and median, it creates perfect symmetry. The perpendicular line through the midpoint acts like a mirror, making $AB = AC$.

Q: What if AD was just a height but not a median?

If AD was only a height, the triangle could be any type - right-angled, scalene, or isosceles. The special property comes from being both height and median simultaneously.

Q: How can I remember this property?

Think of it as "double duty = double symmetry". When one line does two geometric jobs, it creates the symmetry that makes two sides equal!

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Determine which type of triangle is given

00:03 AD is the height of the triangle according to the given data

00:11 AD is a median in the triangle, the median bisects the side

00:17 A triangle where the altitude is also a median must be an isosceles triangle

00:20 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

Look at the triangle below.

AD is the height and median.

What type of triangle is it?

2

Step-by-step solution

We are given a triangle ABC with a vertex A, and it is noted that AD is both the height from A to the base BC and the median of the triangle. Let's analyze the situation:

Step 1: Since AD is the height, it is perpendicular to the base BC. This tells us that the line AD forms a right angle with BC.
Step 2: Being a median indicates that AD also bisects BC, meaning B and C are equidistant from D. Therefore, $BD = DC$ .
Step 3: The dual role of AD — both a height and a median — is critical. In triangles, if a line segment from a vertex also serves as both a height and a median, the triangle is isosceles. This is due to the symmetry introduced by these properties.
Step 4: Correctly, this means two sides of triangle ABC must be equal. Specifically, since AD meets BC at its midpoint and is also perpendicular, this symmetry means side AB is equal to side AC.

Thus, triangle ABC must be isosceles. This perfectly fits the conventional definition where two sides of the triangle are equal and aligns with the properties of a line being both a height and a median.

Therefore, the solution to the problem is Isosceles.

3

Final Answer

Isosceles

Key Points to Remember

Essential concepts to master this topic

Key Property: When height equals median, triangle must be isosceles
Recognition: AD perpendicular to BC and BD = DC creates symmetry
Verification: Check that AB = AC when line serves both roles ✓

Common Mistakes

Avoid these frequent errors

Confusing height and median definitions
Don't think any line from vertex to opposite side is both height and median = wrong triangle type! Height must be perpendicular, median must bisect the side. Always check both conditions: AD ⊥ BC AND BD = DC.

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 height and a median?

+

A height is perpendicular to the opposite side, while a median connects a vertex to the midpoint of the opposite side. When the same line does both jobs, something special happens!

Why does having both properties make it isosceles?

+

When AD is both height and median, it creates perfect symmetry. The perpendicular line through the midpoint acts like a mirror, making $AB = AC$ .

Could this triangle be equilateral instead?

+

An equilateral triangle is isosceles (all sides equal means two sides are equal), but we only know two sides are equal from the given information. We can't conclude all three sides are equal.

What if AD was just a height but not a median?

+

If AD was only a height, the triangle could be any type - right-angled, scalene, or isosceles. The special property comes from being both height and median simultaneously.

How can I remember this property?

+

Think of it as "double duty = double symmetry". When one line does two geometric jobs, it creates the symmetry that makes two sides equal!

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