Ch. 17 Geometry Flashcards
How do you name a line, line segment, and ray that goes from point A to point B?
Line AB or BA (it extends beyond both endpoints)
Line segment AB or BA (it ends at both endpoints)
Ray AB ONLY, cannot be ray BA (it starts/ends at A & extends beyond B)
What is a vertex?
The common endpoint of two rays (vertex of the angle)
Acute, obtuse, right and straight angle measurements
Acute: greater than 0 but less than 90
Right: exactly 90
Obtuse: greater than 90 but less than 180
Straight: exactly 180
What’s a supplementary angle?
Their measures add up to 180
if a + b = 180, then a & b are supplementary
Sum of exterior angles of ANY polygon
360
Area of a triangle
A = 1/2 b h
b & h are always perpendicular
Triangle inequality theorem
A + B > C
A - B < C
Scalene, isosceles, equilateral triangle definitions
Scalene = all three sides (and angles) are different lengths
Isosceles = two sides (and angles) are the same length
Equilateral = all three sides are the same length and all three angles are same measure (60)
Acute, obtuse, and right triangle definitions
Acute = all three angles are less than 90 degrees
Obtuse = one angle is greater than 90 degrees
Right = one angle is exactly 90 degrees
What is the largest an angle can measure in a triangle?
All angles must be less than 180
Converse pythagorean theorem
Given any triangle with sides A, B, and C, if C^2 = A^2 + B^2, then the angle opposite C must measure 90 degrees, and thus the triangle must be a right triangle.
Pythagorean triples
3-4-5 and 5-12-13 are most common (remember hypotenuse MUST be the longest side)
Lookout for multiples:
6,8,10
9,12,15
Isosceles Right Triangle
Two equal sides (legs) & a right angle.
Ratio of sides is x: x: x√2
The diagnols of a square cut the square into what?
Two isosceles right triangles
Sides = x, Diagnol = x√2
30-60-90 Triangle
Ratio of sides is x (opposite 30 degrees) : 2x (opposite 90 degrees) : x√3 (opposite 60 degrees)
If you cut an equilateral triange in half, what do you get?
Two 30-60-90 Triangles
Area of an equilateral triangle
(s^2 (√3)) / 4
What are similar triangles?
triangles w/ the same shape; one triangle is just an enlargement of the other triangle
What makes triangles similar?
- All three corresponding angles match (note: if two corresponding angles are the same, then the third must be the same)
- Sides have lengths in the same ratio
- An angle of one triangle is the same as the angle in another triangle and the sides surrounding these angles are in the same ratio
What are some signs that will help recognize similar triangles?
- Look for vertical angles (always equal). Then if one more angle matches, the triangles are similar (b/c third angle must match).
- Triangles w/i each other. If non-overlapping sides are parallel, they’re similar (transversals make angles similar)
- Right triangles that share common angles
What is a parallelogram? Are any sides/ angles equal?
Quadrilateral with two pairs of parallel sides.
Opposite sides & opposite angles are equal.
What happens when a diagnol cuts a parallelogram?
Two congruent (identical) triangles are created
What is the total angle measure of two consecutive angles in a parallelogram?
ANY two consecutive angles are supplementary; they sum to 180 degrees
Area of a parallelogram
A = base x height
Remember, base & height are always perpendicular
What is a rectangle?
Parallelogram where all angles are 90 degrees.
So all rules of parallelograms apply (consecutive angles are supplementary; opposite sides/ angles are equal)
Area & perimeter of a rectangle
A = L x W or A = base x height
P = 2L + 2W
Diagnol of a rectangle formula & what does a diagnol create?
D = √(L^2 + W^2)
Two congruent triangles (two diagnols create 4 equal legs inside the rectangle)
*NOTE: vertical angles are obvs equal, BUT adjacent angles that are created by the diagnols are ONLY equal if it’s a square
What is a square?
Special rectangle where all sides are equal; all angles = 90
Diagnol of a square formula & what does a diagnol create?
d = s√2 (same as 45-45-90 triangle)
It creates two 45-45-90 triangles.
Diagnols are all perpendicular, so they create 4 right angles in the center of the square
Area & perimeter of a square
A = s^2
P = 4s
Given a rectangle with a fixed perimeter (e.g., 80 inches), what length/width will produce the greatest area?
Square
If P = 80 inches, then s = 20 will give us greatest area (400).
You can try another P = 80 –> L = 48, W = 1 then area = 48
Given a rectangle with a fixed area, which rectangle will produce the smallest perimeter?
A square
What is a trapezoid?
One pair of opposite sides are parallel; the other pair are NOT
The parallel sides are bases; they are never equal lengths
What is an isosceles trapezoid?
Non-parallel sides are equal length
Area of a trapezoid
A = [(Base 1 + base 2) / 2] x height
Average of bases x height
Formula for interior angles of a polygon
let n = number of sides
(n - 2) x 180
What is a regular polygon?
All interior angles are of equal measure and all sides are equal in length
Degree measure of any one interior angle in a regular polygon
[(n - 2) x 180] / n
How can you split a regular hexagon?
Into 6 equilateral triangles
Area of a regular hexagon
= 6 x area of an equilateral triangle
= 6 x [(s^2) √3]/4
= 3 [(s^2) √3]/2
where s = one side of the hexagon
Alternative formula for area of a regular hexagon (involving d = line from one parallel side to the other)
A = 1.5 sd
What is a chord? What is the longest chord?
Line segment that connects any two points on the circle.
If a chord goes thru center, that’s the diameter = longest chord.
What is the total angle measurement of a circle?
360
Area & Circumference of a circle
A = πr^2
C = πd or 2πr
What are the proportions to remember in any circle? (i.e., central angle / 360 = …)
central angle / 360 =
area of sector / area of circle =
arc length / circumference
What is a central angle?
Any angle at the center that’s formed by two radii
When a central angle intercepts an arc, what is the degree measure of the arc?
Central angle = arc
What is an inscribed angle? What is its measure relative to the central angle?
Inscribed angle = Angle formed by two chords of a circle w/ the vertex of the angle on the circumference
Inscribed angle is 1/2 the measure of the arc it intercepts, so inscribed angle = 1/2 measure of central angle.
(Remember - central angle always bigger than inscribed angle)
Inscribed angle & central angle have same endpoints on the circumference (they intercept the same arc)
When is a triangle inscribed in a circle?
When all vertices lie on circumference
If one side of an inscribed triangle = diameter, what does that mean?
It’s a right triangle and hypotenuse = diameter
If a right triangle is inscribed in a circle, what do you know?
The hypotenuse MUST be the diameter
When an equilateral triangle is inscribed in a circle, where is the center of the triangle?
At the center of the circle.
When a circle is inscribed in an equilateral triangle, where do the circle & triangle touch?
They touch at THREE POINTS ONLY:
Each side of the triangle is tangent (perpendicular!!) to the circle.
They touch at the MIDPOINT of the side of the triangle.
This is the LARGEST circle that can fit inside an equilateral triangle.
When a square is inscribed in a circle, what are the square’s diagnols?
Circle’s diameters
(same w/ rectangle –> diagnol = diameter)
When a circle is inscribed in a square, where do the circle & square touch?
At the midpoint of each side;
Each side is tangent to the circle
Circle’s diameter = side of square
This is the largest circle that can fit inside the square
What is a circumscribed square?
When one square is inscribed in another; the OUTSIDE square is called circumscribed
When a square is inscribed in another square, when is the inscribed square largest?
A square is inscribed in another square when ALL FOUR vertices are touching the sides of the outside (circumscribed) square.
The inside square is largest when the four vertices are closest to the outside square’s four vertices. It’s smallest when the four vertices are touching the midpoints of the sides of outside square.
When a regular polygon is inscribed in a circle, the polygon divides the circle’s circumference into what?
arcs of equal length
If a question asks you to calculate the area of a shaded region, how do you do it?
calculate area of entire figure, calculate area of unshaded region, subtract.
Volume of rectangular solid and cube
Volume = length × width × height
Volume of cube = edge^3
Formula for diagnol line in rectangular solid (from corner to opposite corner) and cube
d^2 = L^2 + w^2 + h^2
d = s√3 (for cube), where s is length of one side/ edge
Surface area of a rectangular solid & cube
SA rect = 2(LW) + 2(HW) + 2 (LH)
SA cube = 6s^2
Volume and SA of a right circular cylinder
V = πr^2h
SA = 2πr^2 + 2πrh
