9 Geometry Final

Equation of a circle

(x-h)² + (y-k)² = r² So basically h and k are the ordered pair for the center of the circle and r is obviously the radius. So most of the time you're just going to have to fill these in but other times you might have to count squares to get to the center or to find the radius but don't panic because it's gonna be fine

Simplifying radical expressions

1. Example square root/50 2. Find factors that make up 50 (25 and 2) 3. Because you can multiply those numbers to get the number 50, you can multiply those square roots to get the square root of 50. square root/50 = square root/25 times square root/2 4. Simplify that and get 5 square root/ 2 HAHA YEY WE FINALLY GOT IT

How to find missing sides with trig ratios

1. Fine your angle measurement. In this problem it's 28° 2. Figure out in relation to that angle measurement, if the two side lengths given are the opposite, adjacent, or hypotenuse sides. Use this info to figure out whether you use sine, cosine, or tangent. The two side lengths (19 and x) are the opposite and hypotenuse to 28°. 3. Once you have figured that out, write down this line: sin(28°)=x/19. You put the x over 19 because that's what SOH tells you to do. 4. But wait! There's a fraction! What should we do?? Just multiply the bottom number by both sides of the equal sign. Now your equation reads: 19sin(28°)=x 5. If it ends up that the x is on the bottom, it's ok. Just add the multiply both side by x like always and before typing it into the calculator divide by everything thats not x. Then just solve the problem like normal:)

Parallelogram properties

1. opposite sides are parallel - well duh 2. opposite sides are congruent - also duh 3. opposite angles are congruent - makes sense 4. Consecutive angles add up to 180 - ok also makes sense 5. diagonals bisect each other means what??

Rectangle properties

1. opposite sides are parallel - well duh 2. opposite sides are congruent - also duh 3. opposite angles are congruent - makes sense 4. Consecutive angles add up to 180 - ok also makes sense 5. diagonals bisect each other means what?? also 6. all angles are right angles - that makes my life easier so YES SCORE 7. diagonals are congruent - ok kind of remotely makes sense

Rhombus properties

1. opposite sides are parallel - well duh 2. opposite sides are congruent - also duh 3. opposite angles are congruent - makes sense 4. Consecutive angles add up to 180 - ok also makes sense 5. diagonals bisect each other means what?? also 6. all four sides are congruent - like a square makes sense 7. diagonals are perpendicular what does this mean 8. Diagonals bisect opposite angles - common sense hello??

Number of sides of a regular polygon

360 divided by the angle measurement

Specific exterior angle

360/number of sides

Exterior Angles sum

360°

The volume of a figure is 110.5 ft³. If the dimensions are doubled, what will the new volume be?

A = 2 A³ = 8 8 times 110.5 = 884 Answer = 884

If the dimensions of a figure are doubled, how many times larger will the surface area be?

A = 2 (doubled) A² = 4 Answer = 4 times larger

If the dimensions of a figure are multiplied by 4, how many times larger will the volume be?

A = 4 A³ = 64 Answer = 64 times larger

The surface are of a figure is 62 cm². If the dimensions are multiplied by 5, what will the new surface area be?

A = 5 A² = 25 62 times 25 = 1550 (Multiply the number of times larger by the original number to get answer) Answer = 1550

If the circle is inside of the triangle and touches all edges of the triangle (circumscribed)

All sides are tangent

Shaded Regions

And for this you still have to find all of the shapes, but you subtract the white shape from the shaded shape and there's your answer!

Proving parallelograms in the coordinate plane

Basically just use the distance formula on all sides and if the top and bottom are the same and the side and side are the same then its a parallelogram yay

Line Symmetry

Can be folded and line up exactly like a mirror ex. an equilateral ∆

Point Symmetry

Can be mapped onto it self and looks the same upside down ex. the letter N

Name that quadrilateral

Check side congruency (do distance formula between top corners, bottom corners, top right with bottom right, top left with bottom left,) then diagonal congruency (top left w bottom right, top right w bottom left). Here are the rules to go from there: 1. If all sides are the same and all diagonals are the same then it's a square congrats these are easy 2. If all of the sides are congruent but the diagonals don't work then its a rhombus yay second easiest 3. If only opposite sides are congruent and the diagonals work too..then its a rectangle 4. Opposite sides congruent but not congruent diagonals? parallelogram and we're done

EofC given graph

Count

Exterior Intersections - two tangents

Do the exact same thing as before yay three cheers for less work!

EofC given center and random point

Fill in center and fill in points for x and y

EofC given numbers

Fill them in

Arcs, Angles, and Algebra

HELP HELP HOW IT WAS SO EASY NOW IT MAKES NO SENSE WHAT WHAT HELP

Changing one side's volume or surface area

HELP HOW DO YOU DO THIS

If the tangent thing looks like an ice cream in a cone

If they both are put together so that they're both tangent, they're congruent

Rotational Symmetry

If you turn the figure and it still looks the exact same then it has rotational symmetry congrats. ex. a sun

Finding segment length - type 1

Intersecting chords and secants are inside the circle Formula - ab = cd

Finding segment length - type 3

Intersecting secant and tangent outside of circle Formula - a² = b(b+c)

Finding segment length - type 2

Intersecting secants outside of the circle Formula - a(a+b) = c(c+d)

Reflection

Just find the x or y line they give you and count from there to find where to put the new points. ****x axis means a horizontal line ****y axis means a vertical line

Pythagorean Theorem converse

Kind of a lot of memorizing here. So what you're gonna do is first check to make sure the 2 legs add up to be bigger than the hypotenuse. Then make the equation look like this: c² = a²+b². Square everything you need to square and basically you're just comparing the now squared numbers to c. If c squared is equal to a squared plus b squared then it's a right triangle. If c squared is bigger than a squared plus b squared then it's an obtuse triangle. If c squared is less than a squared plus b squared then the triangle is an acute triangle. its really quite easy and you better not get it wrong

Finding Similar Solids

Make ratios with corresponding sides and simplify. If all are the same number then they're similar and you can be done

On the circle intersections

Now for this kind they kind of look like scales. There's a flat line at the bottom, a circle sitting on top of that, and a line sticking up from the middle of the bottom line pointing in a different direction every time how exciting! So what you're gonna want to do is take the arc measurement (between the bottom line and the moving line) and divide that by 2 to get your angle measurement (made by the bottom line and the moving line)

Triangle proportionality theorem

OK so pretend that this triable is sitting like a regular triangle and theres a line going between the two slanty lines. This theorem says that if the between line is parallel to the bottom sitting line (which it is), then the top division made by this line over the bottom division is equal to the same thing on the other side

Dilations on a graph

Oh these are easy - you better get 100% on this section Avery so help me. So basically the problem is gonna give you a few points and their ordered pairs and you're just gonna multiply them by the scale factor (shown as k)

Congruent Chords and Arcs

Ok so heres what we know: 1. Two chords (like diameters but not through the entire circle) are congruent if their arcs are congruent. ok that makes sense. they also have to equidistant from the circle. that sounds like something that I'm gonna eyeball and get wrong HA CANNOT WAIT 2. If a diameter is perpendicular to a chord, then it bisects the chord and its arc. sounds reasonable

Interior intersections

Ok so there's this circle with two lines with arrows on both ends intersecting in the circle and going through the circle outline. This divides the circle into four parts. To find the angle measurement of a part in between two of the sections, you add length of the arcs between the arrows hitting the circle outline and divide them by two. wow that was a lot of explaining.

Inscribed Quadrilaterals

Opposite angles are supplementary (add up to 180)

Comparing Volume and Surface Areas

Scale Factor = a:b Surface Area = a²:b² Volume = a³:b³ 1. Make ratio, simplify for scale factor, square for surface area ratio, and cube for volume. It's really very simple and I don't understand how you didn't get 100 on this test Avery. For the pictures just figure out which one you have and either square it or cube it or opposite cube it ok? ok

Non isosceles trapezoids

So a non-isosceles trapezoid is basically this picture but ending with the dotted line. Here are the properties of this thing: 1. The lines on the top and bottom are parallel 2.

Area of a sector and how to do it

So basically just multiply the arc degree measure by pi and the radius squared, then divide the whole thing by 360 the end have a nice day.

Tangents - only a tangent if perpendicular to radius - regular tangent property

So basically the whole deal here is how you use the pythagorean theorem to either see if a line is tangent to a circle or or to find x. When you're doing the first thing just do the Pythagorean theorem and if they are the same number then you're good. When doing the latter, do the Pythagorean theorem again to find the missing length but DO NOT FORGET TO DOUBLE IT IF THE CIRCLE CUTS THROUGH THE LINE OK OK

Inverse trig ratios how?`

So basically this is when the angle isn't a real number, but x instead. So what you're gonna want to do is treat it like a regular problem in the beginning. Set it up like this: (this is an example prob, just trust the numbers) sin(x°)=15/24. But instead of just looking confused at this point, you're gonna press the buttons on the calculator so that instead of sin you get sin-1! How exciting. Now you're just going to set it up like this: x=sin-1(15/24) and continue the problem like before!

Composite Figures

So basically you just need to find the areas of all of the shapes in the giant figure and add them together

Central angles and arc measurements

So central angles are what makes a sector basically. When you add all of these together you'll get 360 obviously. The arc you get by having one of these is equal to the angle you get having one of these.

Trigonometric ratio example

So if I was trying to find the tangent (TOA) of x, I would put the opposite number (5) over the adjacent number (7). That would make the trigonometric ratio be 5/7 the end

How to find an interior angle of a regular polygon

So since all of the sides are =, all you have to do here is divide the sum of all the angles by the number of sides

Rotations on a graph

So the numbers you get for this have a formula yay!! So basically here they are: 1. rotating it 90° makes you put the y first, switch the sign, put the x second and keep it the same 2. rotating it 180° makes you keep the x and y in the same place but switching both signs 3. rotating it 270° makes you flip the x and y and switch the x's sign

45-45-90 ∆ remembering things

So these are right triangles with two legs and a hypotenuse. Both of the legs are the exact same length and are both defined by x. To find the hypotenuse, you need to write square root of 2 and put the x on the outside of the square root. **REMEMBER BC THERE ARE TWO LEGS THAT ARE THE SAME LENGTH**

Translation

So these problems will basically just give you this formula (x.y) -> (x+h, y+k) and in the h and k spots are gonna be numbers telling you where and how many blocks you're supposed to move the x and y ordered pairs of your shape`

30-60-90 ∆ remembering things

So these triangles have three different sized sides. The shortest side is always x. The longer leg is always x square root/3 ** REMEMBER BC THERE ARE THREE DIFFERENT SIZED LEGS**, and the hypotenuse is just x times 2.

Midsegment of a trapezoid

So this is like when theres a line in the middle of the trapezoid between the two bases. This makes all three now line equal to one another. Also if you're trying to find the middle length measurement, just add the two original bases together and divide by two.

Inscribed angles - situation 2

So this kind is where there's a ∆ inside of the circle and the longest side of the ∆ is the diameter of the circle. This situation makes it a right triangle. The angle opposite of the hypotenuse is 90 degrees by the way

If multiplied by a scale factor of A

Surface area = a² Volume = a³

Inscribed angles - situation 3

These are really complicated and I don't really get them but the point is that both of the angles equal each other I guess? HELP STILL NEEDED

Isosceles trapezoids

These have the same properties as non isosceles trapezoids but also these 1. Non parallel sides are congruent (the two tilty sides) 2. Lines from opposite corners (the diagonals are congruent) 3. The two angles made by the base line and the two slanty sides are congruent 4. opposite angles are supplementary REMEMBER TAPPING DIAGONALS, SIDE/SIDE, BOTTOM ANGLE/BOTTOM ANGLE

Angle of Elevation

This is a triangle made from someone looking straight across and then up - like a person and a lighthouse next to one another. Use SOHCAHTOA to find answers

Exterior Intersections - one secant and one tangent

This is basically the same - just take the bigger arc, subtract the smaller arc, and divide it by two

SOHCAHTOA

This is just basic trig ok get it right

Exterior Intersections - two secants

This is when theres a circle and there's a piece of pie without the curly crust side sitting on top of the circle. What you're gonna want to do is take the arc measurement from between when're the curly crust would have been, and subtract the arc measurement closer to the pie tip. Then you just divide that by two and you're done

Inscribed angles - situation 1

Ugh these are annoying. So these look like the two secant thing we did except it's completely inside the circle, and you kind of do the same thing. Theres three situations for this. Situation 1 - to find the degree measurement you're just going to divide the arc that's where the curly crust would be by 2. easy as pie haha get it

Angle of Depression

When someone is higher than something else and they look across and down - like a lifeguard in a stand watching the swimmers. Use SOHCAHTOA again here too

Arc length formula

Yeah so just fill this in and you'll be fine

Pythagorean theorem

a²+b²=c²

Circle Circumference

or

How to find the sum of the interior angles

sum = (number of sides minus 2) times 180

Graph quadrant

top left - 2 top right -1 bottom left -3 bottom right -4