Pythagorean Theorem: Finding Ladder Length with 9m Height and 12m Base Distance

Q: Why can't I just add the height and base distance?

Because the ladder forms the hypotenuse of a right triangle! Adding sides gives you the perimeter, not the longest side. You need the Pythagorean theorem: $a^2 + b^2 = c^2$.

Q: How do I know which sides are the legs and which is the hypotenuse?

The hypotenuse is always the longest side, opposite the right angle. In ladder problems, it's the ladder itself. The legs are the height on the wall and distance from the wall.

Q: What if I get a decimal when taking the square root?

Many real-world problems have exact square roots like this one ($\sqrt{225} = 15$). If you get a decimal, round appropriately for the context - usually to one decimal place for measurements.

Q: Can I use this formula for any ladder problem?

Yes! As long as the ladder, wall, and ground form a right triangle. The formula $a^2 + b^2 = c^2$ works for any right triangle where you know two sides.

Q: Why is √225 = 15 and not some other number?

Because $15 \times 15 = 225$! The square root asks "what number times itself gives 225?" You can verify: $15^2 = 15 \times 15 = 225$.

Right Triangle Applications with Ground Distance

A ladder leans against a wall, meeting the wall at a height of 9 meters. The base of the ladder is 12 meters from the wall. How long is the ladder?

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

A ladder leans against a wall, meeting the wall at a height of 9 meters. The base of the ladder is 12 meters from the wall. How long is the ladder?

Step-by-step solution

To find the length of the ladder (hypotenuse), use the Pythagorean theorem: $a^2 + b^2 = c^2$ .

Given: $a = 9$ meters, $b = 12$ meters.

Substitute the known values into the equation: $9^2 + 12^2 = c^2$ .

Calculate: $81 + 144 = c^2$ .

Simplify: $225 = c^2$ .

Find $c$ : $c = \sqrt{225}$ .

Therefore, the length of the ladder is $15$ meters.

Final Answer

15 meters

Key Points to Remember

Essential concepts to master this topic

Rule: Use Pythagorean theorem for right triangle problems
Technique: Calculate $9^2 + 12^2 = 81 + 144 = 225$
Check: Verify $\sqrt{225} = 15$ and $15^2 = 225$ ✓

Common Mistakes

Avoid these frequent errors

Adding the sides instead of using squares
Don't just add 9 + 12 = 21 meters! This ignores the right triangle relationship and gives a completely wrong answer. Always square both legs first, add them, then take the square root.

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

Why can't I just add the height and base distance?

Because the ladder forms the hypotenuse of a right triangle! Adding sides gives you the perimeter, not the longest side. You need the Pythagorean theorem: $a^2 + b^2 = c^2$ .

How do I know which sides are the legs and which is the hypotenuse?

The hypotenuse is always the longest side, opposite the right angle. In ladder problems, it's the ladder itself. The legs are the height on the wall and distance from the wall.

What if I get a decimal when taking the square root?

Many real-world problems have exact square roots like this one ( $\sqrt{225} = 15$ ). If you get a decimal, round appropriately for the context - usually to one decimal place for measurements.

Can I use this formula for any ladder problem?

Yes! As long as the ladder, wall, and ground form a right triangle. The formula $a^2 + b^2 = c^2$ works for any right triangle where you know two sides.

Why is √225 = 15 and not some other number?

Because $15 \times 15 = 225$ ! The square root asks "what number times itself gives 225?" You can verify: $15^2 = 15 \times 15 = 225$ .

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