Right Triangle Problem: Finding Hypotenuse from 6-Unit Median

Q: What exactly is a median in a triangle?

A median is a line segment from any vertex to the midpoint of the opposite side. In right triangles, the median from the right angle to the hypotenuse has a special property!

Q: Does this rule work for all triangles or just right triangles?

This only works for right triangles! In other triangle types, the median formula is much more complex. Always check that you have a right triangle first.

Q: How can I remember this formula?

Think: "Median splits in half, hypotenuse splits in half" - the median length equals exactly half the hypotenuse length: $ m = \frac{c}{2} $

Q: Can I use this to check if a triangle is a right triangle?

Yes! If the median to one side equals half that side's length, then the triangle is a right triangle with the right angle opposite that side.

Right Triangle Medians with Hypotenuse Length

Imagine a right triangle.

The length from the median to the hypotenuse is 6.

What is the length of the hypotenuse?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:08 First, let's find the length of the hypotenuse, A C.

00:13 We have a right triangle based on the given information.

00:18 Here, B D is a median, which means it splits the side equally.

00:23 Now, let's talk about the median to the hypotenuse.

00:27 In a right triangle, the median to the hypotenuse is half the length of the hypotenuse.

00:38 Substitute B D's value and solve for A C.

00:50 And that's how we find the solution!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Imagine a right triangle.

The length from the median to the hypotenuse is 6.

What is the length of the hypotenuse?

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Identify the key formula for medians in right triangles.
Step 2: Apply this formula to find the hypotenuse length.
Step 3: Verify that the result matches one of the given choices.

Now, let's work through each step:

Step 1: In a right triangle, the median from the right angle vertex to the hypotenuse equals half of the hypotenuse length. That is, if the length of the median is denoted as $m = \frac{c}{2}$ , where $c$ is the hypotenuse length.

Step 2: The given length of the median is 6. Thus, we have:

$6 = \frac{c}{2}$

Multiplying both sides by 2 gives:

$c = 2 \times 6 = 12$

Step 3: The calculated hypotenuse length is 12. We check the answer options provided and find that 12 is indeed one of the given choices.

Therefore, the solution to the problem is $c = 12$ .

Final Answer

Key Points to Remember

Essential concepts to master this topic

Rule: Median to hypotenuse equals half the hypotenuse length
Technique: If median = 6, then hypotenuse = 2 × 6 = 12
Check: Half of 12 equals 6, confirming our median length ✓

Common Mistakes

Avoid these frequent errors

Confusing median with altitude or angle bisector
Don't use altitude formulas like h = ab/c for median problems = wrong geometry concept! Altitude creates right angles while median connects to midpoint. Always use the median formula: $m = \frac{c}{2}$ for right triangles.

Practice Quiz

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Can a triangle have a right angle?

FAQ

Everything you need to know about this question

What exactly is a median in a triangle?

A median is a line segment from any vertex to the midpoint of the opposite side. In right triangles, the median from the right angle to the hypotenuse has a special property!

Why does the median to the hypotenuse equal half the hypotenuse?

This is a special property of right triangles! When you draw a median from the right angle vertex to the hypotenuse, it creates two equal parts, each half the hypotenuse length.

Does this rule work for all triangles or just right triangles?

This only works for right triangles! In other triangle types, the median formula is much more complex. Always check that you have a right triangle first.

How can I remember this formula?

Think: "Median splits in half, hypotenuse splits in half" - the median length equals exactly half the hypotenuse length: $m = \frac{c}{2}$

What if I'm given the hypotenuse and need to find the median?

Just divide by 2! If the hypotenuse is 20, then the median is $\frac{20}{2} = 10$ . The relationship works both ways.

Can I use this to check if a triangle is a right triangle?

Yes! If the median to one side equals half that side's length, then the triangle is a right triangle with the right angle opposite that side.

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