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Home » \|\mathbfu_\perp\|^2 = \|\mathbfu\|^2 - (\mathbfu \cdot \mathbfv)^2 / \|\mathbfv\|^2 \cdot - Blask

\|\mathbfu_\perp\|^2 = \|\mathbfu\|^2 - (\mathbfu \cdot \mathbfv)^2 / \|\mathbfv\|^2 \cdot - Blask

📅 March 10, 2026 👤 scraface

Mar 10, 2026

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📌 However, using the inradius formula, we can express the triangle’s area as:

📌 A_{\text{triangle}} = r \cdot s \quad \text{where } s = \frac{a + b + h}{2}

📌 So the ratio becomes:

📌 \frac{A_{\text{circle}}}{A_{\text{triangle}}} = \frac{\pi r^2}{r s} = \frac{\pi r}{s}

📌 Substituting $ s = \frac{a + b + h}{2} $, the final expression for the ratio is:

📌 \frac{\pi r}{\frac{a + b + h}{2}} = \frac{2\pi r}{a + b + h}

📌 Final answer: $ \boxed{\frac{2\pi r}{a + b + h}} $Question: A robot builder has two gears with 12 and 18 teeth. What is the least common multiple of their teeth counts to realign them?

📌 Solution: To find the least common multiple (LCM) of 12 and 18, factorize both: $12 = 2^2 \cdot 3$ and $18 = 2 \cdot 3^2$. The LCM takes the highest powers of all primes: $2^2 \cdot 3^2 = 4 \cdot 9 = 36$. Thus, the LCM is $\boxed{36}$.

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