Find the Median to Hypotenuse: Right Triangle with Side Length 10

Q: What exactly is a median to the hypotenuse?

A median is a line segment from a vertex to the midpoint of the opposite side. The median to the hypotenuse goes from the right angle vertex to the middle of the hypotenuse.

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

The hypotenuse is always the longest side and sits opposite the right angle (the 90° angle). In this diagram, it's side AC with length 10.

Q: Does this formula work for all triangles?

No! The formula $ m = \frac{c}{2} $ only works for the median to the hypotenuse in right triangles. Other triangles need different median formulas.

Median Properties with Right Triangle Geometry

Given the right triangle

Given the size of the string in the triangle is 10.

Find the size of the median to the hypotenuse.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:07 Let's figure out the length of BD.

00:10 Here's what we know. According to the data, this is the side length we have.

00:20 BD is the median. A median cuts the side it's drawn to in half.

00:29 Look, it's a right triangle in the example we have.

00:33 In right triangles, the median to the hypotenuse is half as long as the hypotenuse.

00:41 Now, let's plug in the AC value and solve for BD.

00:51 And there you have it. That's the solution!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Given the right triangle

Given the size of the string in the triangle is 10.

Find the size of the median to the hypotenuse.

Step-by-step solution

To find the length of the median to the hypotenuse in a right triangle, we will follow these steps:

Step 1: Identify the length of the hypotenuse, $c$ , which is given as 10 units.
Step 2: Use the formula for the median to the hypotenuse: $m = \frac{c}{2}$ , where $c$ is the hypotenuse.
Step 3: Substitute the value of $c$ into the formula to find $m$ .

Now, let's perform the calculations:
Since the hypotenuse $c = 10$ , substituting $10$ into the formula gives:

$m = \frac{10}{2} = 5$

Thus, the length of the median to the hypotenuse is $5$ units.

Final Answer

Key Points to Remember

Essential concepts to master this topic

Rule: Median to hypotenuse equals half the hypotenuse length
Technique: Use formula $m = \frac{c}{2}$ where c = 10 gives m = 5
Check: Verify D is midpoint of AC and BD = 5 units ✓

Common Mistakes

Avoid these frequent errors

Using the wrong triangle side for the calculation
Don't use the leg length (10) as if it's the median = gives answer 5 by coincidence! This confuses which measurement is which. Always identify that 10 is the hypotenuse length, then apply the median formula.

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

What exactly is a median to the hypotenuse?

A median is a line segment from a vertex to the midpoint of the opposite side. The median to the hypotenuse goes from the right angle vertex to the middle of the hypotenuse.

Why is the median always half the hypotenuse in right triangles?

This is a special property of right triangles! When you draw a median from the right angle to the hypotenuse, it creates an isosceles triangle where the median equals the two equal sides.

How do I know which side is the hypotenuse?

The hypotenuse is always the longest side and sits opposite the right angle (the 90° angle). In this diagram, it's side AC with length 10.

What if I calculated the median as 10 instead of 5?

You likely used the hypotenuse length directly instead of dividing by 2. Remember: median = $\frac{\text{hypotenuse}}{2} = \frac{10}{2} = 5$

Does this formula work for all triangles?

No! The formula $m = \frac{c}{2}$ only works for the median to the hypotenuse in right triangles. Other triangles need different median formulas.

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