Triangle ABC Triangle FDE are similar. What is the similarity ratio from triangle ABC to FDE?

Bergen
2021-07-31
Answered

Triangle ABC Triangle FDE are similar. What is the similarity ratio from triangle ABC to FDE?

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Clara Reese

Answered 2021-08-01
Author has **120** answers

Step 1

The similarity ratio of triangle ABC and FDE is

Step 2

Given:

$\mathrm{\u25b3}ABC\stackrel{\sim}{=}\mathrm{\u25b3}FDC$

If two triangles are similar AB=FD, BC=DE, AC=FE

$\therefore$ similarity ratio

$\frac{AB}{FD}=\frac{4}{12}=\frac{1}{3}$

AB:DF=1:3

The similarity ratio of triangle ABC and FDE is

Step 2

Given:

If two triangles are similar AB=FD, BC=DE, AC=FE

AB:DF=1:3

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$\begin{array}{r}g(y,x)={\int}_{0}^{x}f(t,y)dt\end{array}$

I am trying to maximize $\underset{y\in S}{max}g(y,x)$. Using Fatou's lemma we have that

$\begin{array}{r}\underset{y\in S}{max}g(y,x)={\int}_{0}^{x}f(t,y)dt\le {\int}_{0}^{x}\underset{y\in S}{max}f(t,y)dt\end{array}$

I also have that $\underset{y\in S}{max}f(t,y)\le h(t)$ where ${\int}_{0}^{x}h(t)<\mathrm{\infty}$

My question when does the last inequality hold with equally?

What do I have to assume about $f(y,x)$ and $g(x,y)$ for equality to hold?

Can I apply dominated convergence theorem here?