euph0ria

since PQ and JL are parallel, we can determine that triangles KJL and KPQ are similar. 

now, we can use the similar triangle ratios to help us find x:

\(\frac{x-6}{x-6+3} = \frac{x-6}{x-3}\)

and we also have

\(\frac{x}{x+5}\)

we know that these two equal each other. so we get an equation:

\(\frac{x-6}{x-3}= \frac{x}{x+5}\)

let's cross multiply, and then continue simplifying -

\((x-6)(x+5) = (x-3)x\)

\(x^2-x-30 = x^2-3x\)

\(2x = 30\)

\(x=15\), which is option D

as you can see, the equation ended up simplifying quite nicely!

hope this helped! please let me know if you are still confused :)

the shaded area is essentially the area of the trapezoid minus the area of that semi-circle.

the area of the trapezoid is 

\(\frac{50+80}{2} \cdot (20 \cdot 2)\)

= \(\frac{130}{2} \cdot 40\)

= \(65 \cdot 40\)

= \(2600\)

the area of the semi-circle is 

\(\frac{20^2 \cdot \pi}{2}\)

= \(\frac{400 \cdot \pi}{2}\)

= \(200 \pi\)

so, the area of the shaded region is \(2600 - 200\pi\).

hope this helped! please let me know if you have any questions 

i'm not exactly sure how to explain these, but i believe number 3 is the answer.

here is a site that describes the properties of a parallelogram:

https://www.mathplanet.com/education/geometry/quadrilaterals/properties-of-parallelograms

if you have any further questions on why 1, 2, and 4 are not properties of a parallelogram, feel free to ask! 

hope this helped!

we can use Pythagorean's theorem for this one. let's just call x the missing side length.

21^2 + 28^2 = x^2

441 + 784 = x^2

1225 = x^2

x = 35

so the missing side length is 35. 

by the way, the Pythagorean's theorem is \(a^2+b^2=c^2\). 

hope this helped! please let me know if you are still confused :)

since a square has 4 equal sides, the perimeter would be 4 * (side length of the square). 

so, each side length of the square would be \(\frac{80-64y}{4}\). 

80 / 4 = 20 and 64y / 4 = 16y

each side length of the square is 20 - 16y, or the third option.

it seems that you are asking many questions about factoring simple expressions. here is a site that may help:

https://www.factoring-polynomials.com/factoring-2.htm

i have mentioned this before, but Symbolab's simplifying calculators are helpful as well, in providing step-by-step solutions and tips. feel free to explore!

hope this helped! sorry if my explanations weren't that great. a quick search online will also always provide you with many resources and tips! 

i will first separate 12b into two different factors: 12 and b - we will factor these out one at a time (hopefully this step makes sense). this way it will be easier to understand, and solve. 

first, let us factor out the 12 from the expression. 60 / 12 = 5, and 72 / 12 = 6. 

this would be the current expression: 12(5ab - 6b). now, let's disregard the 12 factor for a moment, so we can factor out the b in the other factor. (sorry if the wording here is a bit strange)

5ab / b = 5a, and -6b / b = -6. 

here is the resulting expression (make sure to remember the 12 we factored out at the beginning!):

12b(5a - 6), which is the second option.

hope this helped! please let me know if you have any questions :)

let's start by converting all of these into grams. this way, it will be easier to put in order. 

1 kilogram = 1000 grams.

so, 7 kg would be 7000 grams, and 3 kg would be 3000 grams.

now, we have:

7000 grams

600 grams

100 grams

3000 grams

we can now order these from least to greatest, because they are all in the same unit.

the final order would be:

100 grams

600 grams

3000 grams

7000 grams

now we have to convert it back, to suit the original problem.

100 grams

600 grams

3 kg

7 kg

and we're done! hope this helped 

angle 1 would just be 24, because all the triangles inside the rhombus are congruent.

angle 2 would be 90, because the diagonals of a rhombus are perpendicular (if you would like proof, i believe Khan Academy has a video on that). 

angle 3 would be 180 - (90 + 24) = 180 - 114 = 66.

hope this helped! please let me know if you are still confused or have any questions :)

to expand this polynomial, we should multiply the factors in order from left to right. hopefully you can follow this process - 

5w + 5*4 + w^2 + 4w.

to simplify, we combine the like terms, to get:

w^2 + 9w + 20

hope this helped! sorry if this wasn't too clear. 

here are a few websites that may help you understand this better:

https://www.brainfuse.com/jsp/alc/resource.jsp?s=gre&c=35257&cc=108827#:~:text=A%20polynomial%20is%20expanded%20if%20no%20variable%20appears,3x(4%20-%20x)%20%2B%20(2x%20-%205)%202 

also try out Symbolab's expansion calculator. it provides a step-by-step expansion of polynomial expressions as well.

if you have some time, give these a try! Google searches are also useful to help you learn more too!