S3 Geometry and Mensuration Flashcards
What are the features of an angle bisector of a triangle?
- Every triangle has 3 angle bisectors.
- An angle bisector must start from a vertex of a triangle.
- All angle bisectors lie inside the triangle.`
What are the features of a median of a triangle?
- Every triangle has 3 medians.
- A median must start from a vertex, and end at the mid-point of the opposite side.
- All medians lie inside the triangle.
What are the features of a perpendicular bisector of a triangle?
- Every triangle has 3 perpendicular bisectors.
- A perpendicular bisector may not pass through a vertex.
- If a perpendicular bisector passes through a vertex, the triangle must be isosceles.
What are the features of an altitude of a triangle?
- Every triangle has 3 altitudes.
- An altitude must start from a vertex, and end on the opposite side (or the extended part of the opposite side).
- An altitude may lie inside or outside of the triangle; it may also coincide with a side of the triangle.
Angle bisector property
If a point lies on the angle bisector, then it is equidistant from the two sides of the angle.
Converse of angle bisector property
If a point is equidistant from the two sides of an angle, then it lies on the angle bisector.
Angle bisector theorem
In triangle ABC, if AD is an angle bisector of angle BAC, then AB/AC = BD/DC.
Property of perpendicular bisector
If a point lies on the perpendicular bisector of a line segment, then is it equidistant from the two end points of the line segment.
Converse of perpendicular bisector property
If a point is equidistant from the two end points of a line segment, then it lies on the perpendicular bisector of the line segment.
Concurrent lines
If more than two lines pass through the same point, they are concurrent lines.
Definition of incentre
The intersection of 3 angle bisectors of a triangle
Features of incentre
- It lies inside the triangle.
- The perpendicular distance between the incentre and the sides of the triangle are equal.
- A circle centred at the incentre can be drawn such that it touches each edge of the triangle at one point. It is the largest circle that can be enclosed by the triangle. (inscribed circle)
Definition of centroid
The intersection of 3 medians of a triangle
Features of centroid
- It lies inside the triangle.
- The centroid divides each median in the ratio 2:1.
- In coordinate geometry, the coordinates of the centroid is the average of the coordinates of the three vertices.
Definition of circumcentre
The intersection of 3 perpendicular bisectors of a triangle
Features of circumcentre
- It may lie inside, on, or outside the triangle. When it lies on the triangle, the triangle must be a right-angled triangle and the circumcentre is located at the mid-point of the hypotenuse.
- The circumcircle of the triangle (a circle through all vertices of the triangle) is centred at the circumcentre.
- The circumcentre is equidistant from the vertices of the triangle.
Definition of orthocentre
The intersection of 3 altitudes of a triangle
Features of orthocentre
The orthocentre may lie inside, on, or outside a triangle. If it lies on the triangle, then the triangle is a right-angled triangle and the orthocentre is located at the vertex opposite the hypotenuse.
Denote the centroid, circumcentre, and orthocentre of a right-angled isosceles triangle as G, C, and H respectively. Find HG:GC.
2:1
C is the mid-point of the hypotenuse of the triangle.
H coincides with the vertex opposite to the hypotenuse of the triangle.
Therefore, HC is a median. HGC is a straight line.
Using the property that the centroid divides the median into ratio 2:1, HG:GC=2:1.
if ABCD is a //gram, then (def. of //gram):
AB//DC, AD//BC
if ABCD is a //gram, then (opp. sides of //gram):
AB=DC and AD=BC
if ABCD is a //gram, then (opp. <s of //gram):
<A = <C and <B = <D
if ABCD is a //gram and E is the intersection of its diagonals, then (diagonals of //gram):
AE=EC and BE=ED
If quadrilateral ABCD satisfies AB//DC and AD//BC, then ABCD is a //gram…
def. of //gram
If quadrilateral ABCD satisfies AB=DC and AD=BC, then ABCD is a //gram…
opp. sides equal
If quadrilateral ABCD satisfies <A = <C and <B = <D, then ABCD is a //gram…
opp. <s equal
If quadrilateral ABCD with intersection of diagonals as E satisfies AE=EC and BE=ED, then ABCD is a //gram…
diags. bisect each other
If quadrilateral ABCD satisfies AB//DC and AB=DC, then ABCD is a //gram…
2 sides equal and //
Primal definition of rectangle
A quadrilateral with 4 equal interior angles
Primal definition of rhombus
A quadrilateral with 4 equal sides
Primal definition of square
A quadrilateral with four equal sides and four equal interior angles
Definition of kite
A quadrilateral with two pairs of equal adjacent sides
Definition of trapezium
A quadrilateral with ONLY one pair of parallel sides
Definition of parallelogram
A quadrilateral with two pairs of parallel sides
(property of rhombus):
- all properties of //grams
- two diagonals are perpendicular to each other
- diagonals bisect interior angles.
(property of rectangle):
- all properties of //grams
- all interior angles are 90 degrees
- diagonals are equal
- diagonals bisect each other into 4 equal line segments.
(property of square):
- All properties of //grams
- all properties of rectangles
- all properties of rhombus
- angle between a side and a diagonal is 45 degrees.
Properties of isosceles trapeziums
- Upper base angles are equal
- Lower base angles are equal
- Diagonals are equal in length
Properties of kite
- Diagonals are perpendicular to each other.
- It has reflectional symmetry.
In triangle ABC, define M and N as the mid-points of AB and AC respectively. Then… (mid-pt thm.)
MN//BC, MN=1/2BC
In triangle ABC, define M and N as the mid-points of AB and AC respectively. Then MN=1/2BC …
(mid-pt thm.)
In triangle ABC, define M and N as the mid-points of AB and AC respectively. Then MN//BC …
(mid-pt thm.)
In triangle ABC, define M as the mid-point of AB and N such that MN//BC. Then… (intercept thm.)
AN=NC
In triangle ABC, define M as the mid-point of AB and N such that MN//BC. Then AN=NC …
(intercept thm.)
In triangle ABC, define M as the mid-point of AB and N such that MN//BC. Prove that MN=1/2BC.
- AN=NC (intercept thm.)
- as AM=MB and AN=NC, MN=1/2BC (mid-pt thm.)
Define three parallel lines AB//CD//EF. If AC=CE, then…
BD=DF (intercept thm.)
Define three parallel segments AB//CD//EF. If AC=CE, prove that BD=DF.
Connect AF. Label the intersection of AF and CD as G.
1. AC=CE, CG//EF: AG=GF (intercept thm.)
2. AG=GF, AB//GD: BD=DF (intercept thm.)
In triangle ABC, define M and N on AB and AC respectively such that MN//BC. Then… (equal ratio theorem)
AM/BM = AN/CN
In triangle ABC, define M and N on AB and AC respectively such that MN//BC, and AM:BM = 2:1. Then AN:CN = 2:1 …
(equal ratio theorem)
Define three parallel segments AB//CD//EF. Then… (equal ratio theorem)
AC/CE = BD/DF
What is the shape of a cross section of a prism/cylinder when it is cut parallel to the base?
It is identical to the base in shape and size.
What is the shape of a cross section of a prism/cylinder when it is cut perpendicular to the base?
A rectangle
What is the shape of a cross section of a pyramid/cone when it is cut parallel to the base?
It is similar to the base
What is the shape of a cross section of a pyramid/cone when it is cut perpendicular to the base?
A triangle
What is the definition of a regular tetrahedron?
A triangular pyramid in which all four faces are equilateral triangles. (all the edges are equal in length)
Points to note when drawing 2-D representations of right pyramids/cones
- Use solid lines to represent the outermost lines and dotted lines to represent auxiliary lines (eg height) and hidden edges.
- Add a height and mark the right angle to show that it is perpendicular to the base.
Right pyramid
Its vertex is vertically above the centroid of its base
Oblique pyramid
Its vertex is not vertically above the centroid of its base
Regular pyramid
Its base is a regular polygon with the same side lengths and interior angles.
It is a right pyramid.
Volume of prism
base area x height
Volume of pyramid
1/3 x base area x height
Slant height of cone
The distance between the vertex and any point on the circumference of the base. All slants heights of a right circular cone have the same length.
Right circular cone
Its foot of the height is the centre of the base.
Oblique cone
Its foot of the height is not the centre of the base.
Volume of cylinder
π r2h
Volume of cylinder
1/3 π r2h
What is a frustum of a pyramid/cone?
A frustum is obtained by cutting a pyramid or come by a plane parallel to the base.
Total surface area of a pyramid
Total area of all lateral faces + base area
Curved surface area of cone
π r l (slant height)
Total surface area of cone
π r l + π r2
Surface area of sphere
4π r2
Total surface area of hemisphere
3π r2
Volume of sphere
4/3π r3
Given similar figures with length ratio x:y, their area ratio is
x2:y2
Given similar solids with length ratio x:y, their volume ratio is
x3:y3
Divide a triangle ABC by segments parallel to BC that trisect AB and AC. Define the parts from the closest to A to the furthest as X, Y, and Z respectively.
S(X):S(Y):S(Z) =
1:3:5
S(X) + S(Y) = 4S(X)
S(X) + S(Y) + S(Z) = 9S(X)
Divide a pyramid with vertex V by planes parallel to its base which trisect the slant heights. Define the parts from the closest to V to the furthest as X, Y, and Z respectively.
V(X):V(Y):V(Z) =
1:7:19
V(X) + V(Y) = 8V(X)
V(X) + V(Y) + V(Z) = 27V(X)
