What do you need to know to prove that two triangles are congruent using the SSS Congruent Postulate if it's given that two pairs of sides are congruent?

tinfoQ
2021-08-06
Answered

What do you need to know to prove that two triangles are congruent using the SSS Congruent Postulate if it's given that two pairs of sides are congruent?

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stuth1

Answered 2021-08-07
Author has **97** answers

The SAS Congruence Postulate proves that triangles are congruent when all three pairs of corresponding sides are congruent. Since we know that two pairs are congruent, information about the third pair being congruent is what's missing.

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I have a function $f=-20p\cdot q+9p+9q$. Player 1 chooses $p$ and player 2 chooses $q$. Both $p$ and $q$ are in the inclusive interval [0,1]. Player 1 wants to maximize $f$ while player 2 wants to minimize $f$.

Player 1 goes first, what is the most optimal value of $p$ he should choose knowing that player 2 will choose a $q$ in response to player 1's choice of $p$?

This seems to be some sort of minimization-maximization problem, but I am unsure how to solve it. I was thinking about approaching this from a calculus perspective by taking the partial derivative of $f$ with respect to $p$, but it doesn't seem I get an intuition by doing this, and it seems that $p$ and $q$ are a function of each other. How should I approach solving this problem analytically?

Player 1 goes first, what is the most optimal value of $p$ he should choose knowing that player 2 will choose a $q$ in response to player 1's choice of $p$?

This seems to be some sort of minimization-maximization problem, but I am unsure how to solve it. I was thinking about approaching this from a calculus perspective by taking the partial derivative of $f$ with respect to $p$, but it doesn't seem I get an intuition by doing this, and it seems that $p$ and $q$ are a function of each other. How should I approach solving this problem analytically?

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Let $A$ be a matrix and $Q$ be an orthogonal matrix such that $A{Q}^{T}$ is symmetric, positive semidefinite. Show that

$||A+Q|{|}_{F}\ge ||A+P|{|}_{F}$

for any orthogonal matrix $P$. Here, $||\cdot |{|}_{F}$ is the Frobenius norm.

$||A+Q|{|}_{F}\ge ||A+P|{|}_{F}$

for any orthogonal matrix $P$. Here, $||\cdot |{|}_{F}$ is the Frobenius norm.