Triangle Median Problem: Finding AC Length When BC = AC

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

A median connects a vertex to the midpoint of the opposite side, while an altitude creates a right angle with the opposite side. In this problem, AD is specifically called a median, so it splits BC into equal parts.

Q: How do I know this triangle is isosceles?

The problem states that BC = AC. When two sides of a triangle are equal, it's called an isosceles triangle. This is given information, not something we need to prove!

Q: Why does the median create equal segments BD and CD?

That's the definition of a median! A median always connects a vertex to the midpoint of the opposite side. Since D is the midpoint of BC, we automatically get $ BD = CD $.

Q: Could AC be a different length than 10?

No! The problem gives us two key facts: AD is a median (so $ BD = CD = 5 $) and $ BC = AC $. Since $ BC = 10 $, we must have $ AC = 10 $.

Q: Do I need to use the Pythagorean theorem here?

Not for this problem! We're using basic triangle properties and the given condition that $ BC = AC $. The Pythagorean theorem would only be needed if we were dealing with right triangles or finding unknown side lengths.

Isosceles Triangle Properties with Median Applications

If AD is the median.

and BC is equal to AC

Determine the length of the side AC.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Calculate the side AC

00:03 AD is a median according to the given information. The median bisects the side

00:08 Insert the value of BD into the formula, according to the given information

00:14 The entire side equals the sum of its parts

00:20 Substitute in the relevant values and proceed to solve to find BC

00:27 The sides are equal according to the given information

00:33 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

If AD is the median.

and BC is equal to AC

Determine the length of the side AC.

Step-by-step solution

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

Step 1: Identify the given information that $AD$ is the median making $BD = CD = 5$ .
Step 2: Use the triangle property that $BC = AC$ as per the problem’s isosceles condition.
Step 3: Calculate $BC = BD + CD = 5 + 5 = 10$ .
Step 4: Since $BC = AC$ , by isosceles property, conclude $AC = 10$ .

Now, let's work through each step:
Step 1: With $AD$ as the median, it divides $BC$ into $BD = 5$ and $CD = 5$ .
Step 2: The condition $BC = AC$ ensures the triangle is isosceles.
Step 3: Calculate $BC$ as $BD + CD = 5 + 5 = 10$ .
Step 4: Since $BC = AC$ , therefore, $AC = 10$ .

Therefore, the length of $AC$ is $10$ .

Final Answer

Key Points to Remember

Essential concepts to master this topic

Median Property: Divides opposite side into two equal segments
Technique: Since $BD = CD = 5$ , calculate $BC = 5 + 5 = 10$
Check: Use isosceles condition: if $BC = AC$ and $BC = 10$ , then $AC = 10$ ✓

Common Mistakes

Avoid these frequent errors

Confusing median with altitude or angle bisector
Don't assume AD creates right angles or equal angles! A median only connects vertex to midpoint of opposite side. This leads to using wrong triangle properties. Always remember: median = midpoint connection, creating equal segments on the base.

Practice Quiz

Test your knowledge with interactive questions

Can a triangle have a right angle?

FAQ

Everything you need to know about this question

What's the difference between a median and an altitude?

A median connects a vertex to the midpoint of the opposite side, while an altitude creates a right angle with the opposite side. In this problem, AD is specifically called a median, so it splits BC into equal parts.

How do I know this triangle is isosceles?

The problem states that BC = AC. When two sides of a triangle are equal, it's called an isosceles triangle. This is given information, not something we need to prove!

Why does the median create equal segments BD and CD?

That's the definition of a median! A median always connects a vertex to the midpoint of the opposite side. Since D is the midpoint of BC, we automatically get $BD = CD$ .

Could AC be a different length than 10?

No! The problem gives us two key facts: AD is a median (so $BD = CD = 5$ ) and $BC = AC$ . Since $BC = 10$ , we must have $AC = 10$ .

Do I need to use the Pythagorean theorem here?

Not for this problem! We're using basic triangle properties and the given condition that $BC = AC$ . The Pythagorean theorem would only be needed if we were dealing with right triangles or finding unknown side lengths.

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