If in $\triangle \mathrm{A B C}$ and $\triangle \mathrm{D E F}$, AB = DE, $\mathrm{m} \angle \mathrm{A B C=\mathrm{m} \angle E D F}$, BC = DF, $\mathrm{A C}=(20 x-10)$, and $\mathrm{E F}=(15 x+15)$, what is the value of x?

$\triangle A B C \sim \triangle F D E$. What is DE?

Given that the length of $\overline{AC}=12$, $\overline{BC}=8$, $\overline{EF}=18$, and $\overline{DF}=6$

(a). Draw trapezoid ABCD. Let E and F be points on $\overleftrightarrow{A B}$ such that $\overrightarrow{C E} \perp \overleftrightarrow{A B}$ and $\overrightarrow{D F} \perp \overleftrightarrow{A B}$.

(b). Prove that CE=DF.

($c$). Let AB=$b_1, CD=b_2$, and CE=DF=h. Prove that the area of trapezoid ABCD is $\frac{h}{2}\left(b_1+b_2\right)$.

If $F(x, y, z)=\left(5 x^2 y / z^3\right), x=t^{3 / 2}+2, y=\ln 4 t$, and $z=e^{3 t}$, find dF/dt in terms of x,y,z, and t.

Let f(x)=x^3-3 x^2-1, $x \geq 2$ Find the value of df^{-1} / dx at the point x=-1=f(3)

BC=EF and AC=DF. Are the triangles congruent?

The triangle given are triangles ABC and DEF. The vertices are at A(-3,2), B(-3,4), C(1,4), D(-7,-2), E(-7,0), and (-3,0).

A. Yes

B. No

True/False. If $\mathbf{f}: \mathbf{R}^{3} \rightarrow \mathbf{R}^{4}$ is differentiable, then Df(x) is a $3 \times 4$ matrix.

Perform the operation : $DF$, where matrices $A$ and $C$ are :

$$D= \begin{bmatrix} 3 & 0\\ 1 & -1 \\ 2 & 5 \end{bmatrix} \ \ \ \ F= \begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix}$$

Find the following ratio $\triangle D E F$ where $m \angle E$ = $90^\circ$ DE = 4, EF= 3, DF = 5.

$$\cos D$$

Find the following ratio $\triangle D E F$ where $m \angle E$ = $90^\circ$ DE = 4, EF= 3, DF = 5.

$$\tan F$$

Suppose that the differentiable function y=f(x) has an inverse and that the graph of f passes through the point (2,4) and has a slope of 1/3 there. Find the value of df^{-1} / dx at x=4.

In $\triangle \mathrm { DEF } ,$ find $\angle \mathrm { F }$ to the nearest degree, if $\mathrm { DE } = 15 \mathrm { cm } , \mathrm { DF } = 18 \mathrm { cm } , \text { and } \angle \mathrm { E } = 90 ^ { \circ }$

Use $\triangle G H J$, where D, E, and F are midpoints of the sides.

If HJ=8x-2 and DF=2x+11, what is HJ?

Find the following ratio $\triangle D E F$ where $m \angle E$ = $90^\circ$ DE = 4, EF= 3, DF = 5.

$$\sin D$$