Triangle Median Problem: When BD = 1/2 AC, Is ABC Right?

Q: How do I know which side is the hypotenuse?

The hypotenuse is always the longest side in a right triangle, and it's opposite the right angle. In this problem, since the median goes to side AC, then AC is the hypotenuse.

Q: What exactly is a median in a triangle?

A median is a line segment that connects a vertex to the midpoint of the opposite side. So BD connects vertex B to point D, which is exactly halfway along side AC.

Q: Why does this median property only work for right triangles?

This is a unique characteristic of right triangles! In any other triangle, the median to the longest side is either longer or shorter than half that side - never exactly half.

Q: Can I use this rule backwards to prove a triangle is right?

Absolutely! This is actually a common proof technique. If you can show that a median equals half the side it goes to, you've proven the triangle is right-angled.

Right Triangle Identification with Median Properties

BD is the median in triangle ABC and is half as long as side AC.

Is ABC a right triangle?

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Step-by-step written solution

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Understand the problem

BD is the median in triangle ABC and is half as long as side AC.

Is ABC a right triangle?

Step-by-step solution

In this problem, we are given that $BD$ is the median of triangle $\triangle ABC$ and that $BD$ is half the length of side $AC$ . We want to determine if $\triangle ABC$ is a right triangle.

By the properties of triangles, if the median ( $BD$ ) from the vertex to the hypotenuse ( $AC$ ) of a triangle is half the length of the hypotenuse ( $AC$ ), then the triangle is right-angled.

The key property here is that in a right triangle, the median to the hypotenuse is half the hypotenuse. This median property is a unique characteristic for right triangles.

Thus, since $BD = \frac{1}{2}AC$ and $BD$ is the median, we conclude that triangle $\triangle ABC$ is indeed a right triangle.

Therefore, the solution to the problem is: Yes, $\triangle ABC$ is a right triangle.

Final Answer

Yes

Key Points to Remember

Essential concepts to master this topic

Median Rule: In right triangles, median to hypotenuse equals half hypotenuse
Technique: If $BD = \frac{1}{2}AC$ and BD is median, triangle is right
Check: Verify D is midpoint of AC and angle at B is 90° ✓

Common Mistakes

Avoid these frequent errors

Confusing median direction with right angle location
Don't assume the median goes from the right angle vertex = wrong identification! The median from vertex B to side AC being half of AC means the right angle is AT vertex B, not at the endpoints. Always remember: median to hypotenuse comes FROM the right angle vertex.

Practice Quiz

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

FAQ

Everything you need to know about this question

How do I know which side is the hypotenuse?

The hypotenuse is always the longest side in a right triangle, and it's opposite the right angle. In this problem, since the median goes to side AC, then AC is the hypotenuse.

What exactly is a median in a triangle?

A median is a line segment that connects a vertex to the midpoint of the opposite side. So BD connects vertex B to point D, which is exactly halfway along side AC.

Why does this median property only work for right triangles?

This is a unique characteristic of right triangles! In any other triangle, the median to the longest side is either longer or shorter than half that side - never exactly half.

Can I use this rule backwards to prove a triangle is right?

Absolutely! This is actually a common proof technique. If you can show that a median equals half the side it goes to, you've proven the triangle is right-angled.

Where exactly is the right angle in triangle ABC?

The right angle is at vertex B! Since the median BD goes from B to the hypotenuse AC, the right angle must be at the vertex where the median starts.

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