Right Triangle Side Length: Find BC Given Sides 5 and 13

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

The hypotenuse is always the longest side in a right triangle and sits opposite the right angle (90° angle). In this problem, side AC = 13 is the hypotenuse because it's the longest side.

Q: Why do we square root at the end?

After solving $ BC^2 = 144 $, we need to find BC itself, not $ BC^2 $. Taking the square root: $ BC = \sqrt{144} = 12 $.

Q: What if I get a decimal or radical answer?

That's completely normal! Not all right triangles have integer side lengths. Leave radicals in simplest form like $ \sqrt{8} = 2\sqrt{2} $ or use decimals when appropriate.

Q: Can I use this formula for any triangle?

No! The Pythagorean Theorem only works for right triangles. You must see a 90° angle (usually marked with a small square) to use this method.

Q: What if both legs are unknown?

You need at least two sides to find the third. With only one side given, there are infinitely many possible right triangles, so the problem cannot be solved.

Pythagorean Theorem with Unknown Leg

Given the triangle ABC, find the length BC

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find BC

00:03 We'll use the Pythagorean theorem in a right triangle

00:11 We'll substitute appropriate values and solve for BC

00:19 Isolate BC

00:32 Take the square root

00:39 And this is the solution to the problem

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Given the triangle ABC, find the length BC

Step-by-step solution

To answer this question, we must know the Pythagorean Theorem

The theorem allows us to calculate the sides of a right triangle.

We identify the sides:

ab = a = 5
bc = b = ?

ac = c = 13

We replace the data in the exercise:

5²+?² = 13²

We swap the sections

?²=13²-5²

?²=169-25

?²=144

?=12

Final Answer

12 cm

Key Points to Remember

Essential concepts to master this topic

Pythagorean Theorem: For right triangles, $a^2 + b^2 = c^2$ where c is hypotenuse
Technique: Rearrange to $b^2 = c^2 - a^2 = 13^2 - 5^2 = 144$
Check: Verify $5^2 + 12^2 = 25 + 144 = 169 = 13^2$ ✓

Common Mistakes

Avoid these frequent errors

Using the wrong side as the hypotenuse
Don't assume the first given side is the hypotenuse = wrong equation setup! The hypotenuse is always the longest side opposite the right angle. Always identify which side is the hypotenuse by looking at the diagram and finding the longest side.

Practice Quiz

Test your knowledge with interactive questions

Consider a right-angled triangle, AB = 8 cm and AC = 6 cm.
Calculate the length of side BC.

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 sits opposite the right angle (90° angle). In this problem, side AC = 13 is the hypotenuse because it's the longest side.

Why do we square root at the end?

After solving $BC^2 = 144$ , we need to find BC itself, not $BC^2$ . Taking the square root: $BC = \sqrt{144} = 12$ .

What if I get a decimal or radical answer?

That's completely normal! Not all right triangles have integer side lengths. Leave radicals in simplest form like $\sqrt{8} = 2\sqrt{2}$ or use decimals when appropriate.

Can I use this formula for any triangle?

No! The Pythagorean Theorem only works for right triangles. You must see a 90° angle (usually marked with a small square) to use this method.

What if both legs are unknown?

You need at least two sides to find the third. With only one side given, there are infinitely many possible right triangles, so the problem cannot be solved.

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Right Triangle Side Length: Find BC Given Sides 5 and 13

Pythagorean Theorem with Unknown Leg

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

How do I know which side is the hypotenuse?

Why do we square root at the end?

What if I get a decimal or radical answer?

Can I use this formula for any triangle?

What if both legs are unknown?

🌟 Unlock Your Math Potential

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