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We know that $\triangle BDF$ and $\triangle ECF$ have the same area, and adding the area of quadrilateral $ABFE$ to each of them shows us that $$\text{(area of $\triangle ADE$)}=\text{(area of $\triangle ABC$)}.$$ Using the area formula for triangles, we can turn the last equation into $$\dfrac{1}{2}\cdot AD\cdot AE = \dfrac{1}{2}\cdot AB\cdot AC.$$ Substituting in the given lengths, we get $$\dfrac{1}{2}\cdot 5\cdot 4 = \dfrac{1}{2}\cdot 3\cdot AC,$$ so $AC=\dfrac{20}{3}$. We compute the length we are interested in as $EC=AC-AE=\dfrac{20}{3}-4=\boxed{\dfrac{8}{3}}$.

No prob

Thank you

draw a rectangle around it and then find the area of the 3 triangles around the gray triangle, then subtract it from the big rectangle and you get 9

1: Use this example to demonstrate: Points M and G are labeled and represent the Mirror and the point where the tree touches the ground.

a: Label points: T at the top of the tree, E at the person’s eye level, and F at the person’s feet

b: Explain how you can create two similar triangles using the AA similarity postulate. You can sketch the triangles separately if you would like (hint: if the person and tree stand up straight, they form 90-degree angles at the ground):

c: If EF = 60 inches, FM = 80 inches, and MG = 200 inches, how tall is the tree?

Hello buyya!

1b.

In optics, angles of incidence are equal to angles of reflection in the case of reflection.

$\angle EMF=\angle TMG$

$\angle EFM=\angle TGM=90^{\circ}$

$ΔMEF\sim ΔMTG$

1c.

$\overline{TG}:\overline{EF}=\overline{GM}:\overline{FM}\\ \overline{TG}:60''=200'':80''\\ \overline{TG}\cdot 80''=60''\cdot 200''\\ \overline{TG}= \dfrac{60''\cdot 200''}{80''}\\ \overline{TG}=150''$

$The\ tree\ is\ 150\ inches\ tall.$

2.

$\angle DBC=\angle GBA\ (vertex\ angle )$

$\angle DCB=\angle GAB=90^{\circ}$

$ΔDBC\sim ΔGBA$

$\overline{AG}:\overline{CD}=\overline{BA}:\overline{BC}\\ \overline{AG}:3'=15':4'\\ 4'\cdot \overline{AG}=15'\cdot 3'\\ \overline{AG}=\dfrac{15'\cdot 3'}{4'}$

$\overline{AG}=11\frac{1}{4}'$

$The\ river\ is\ 11\frac{1}{4}\ feet\ wide.$

  !

Peter can make 156 different amounts.

y = -2x + b

0 = -2(-5) +b     b = -10

Are you sure that is correct?    I get  3/4 ............

from  3 out of 9    to    4 out of 10        to   9 out of 15       five more in a row    

x = saves    y = attempts

.33333y = x     and     .4(y+1) = x+1    which is   x = .4y-.6    sub this into the forst equation

 .33333y = .4y-.6

results in y = 9      so   x = 6        add 1 attempt and 1 save to get    4 out of 10      

    then    add 5 more to get 9 out of 15 = 60 %

"During a football/soccer match, a goalkeepers save rate was 33 1/3%. .After saving the next shot, it rose to 40%. How many more consecutive shots on target must be saved to raise the save rate to 60%?"

Hello! I'll give you a hint: use the PEMDAS method. This stands for Parentheses, Exponents, Multiplication and Division from Left to Right and Addition and Subtraction from Left to Right. Hopefully, this makes sense! - BasicMaths

Translation:

Привет! Подскажу: используйте метод PEMDAS. Это означает круглые скобки, экспоненты, умножение и деление слева направо, а также сложение и вычитание слева направо. Надеюсь, в этом есть смысл! - Основы математики

Asking my teacher....I will respond ASAP when I get the answer.

nvm, its 2/3

50 (tan 35 - tan 28.5) = 7.86 ft

Because CD is parallel to AB, ∠DCP=∠CAB because alternate interior angles are congruent. Similarly, ∠CDP=∠DBA. This means that triangle CDP and triangle BPA are similar by the AA postulate. Since we know the area and the base of triangle CDP, we can calculate the height by saying area=base*height/2. 5=2*height/2. Thus, the height is 5. These two triangles have a scale factor of 2/4. Therefore, we can find the height of the second triangle by proportions. 5/x=2/4. By cross multiplying, we get x=10. To find the height of the whole entire trapezoid, we add the height of the first triangle with the height of the second triangle. 5+10=15. Next, by using the area of trapezoids formula A=h(a+b)/2, we get 15(6)/2=45.

31 different amounts

175 + 8x ≥ 425

By the triangle inequality, the minimum of the remaining side is 80 - 25 = 55.

revenue = (40 000 - 100x)(200 + 10x)                     where x = the number of 10 dollar inreases

   revenue =   - 1000x^2 + 380 000 x +   8 000 000    simplify a bit (to make it easier to work)

                                - x^2 + 380 x  + 8000

Max will be at   - b/2a =   - 380 /(2(-1)) = 190 = x

190 x 10 = 1900       increase              1900 + 200 = 2100 dollars for max revenue   (Yikes!    That doesn't seem correct....check my math!)

triangle side length rule:

7  + 11 = 18            7