June Program Math P1 (outdated)

How to prove that: The line drawn from the centre of a circle perpendicular to a chord bisects the chord. (𝒍𝒊𝒏𝒆 𝒇𝒓𝒐𝒎 𝒄𝒆𝒏𝒕𝒓𝒆 ⊥ 𝒕𝒐 𝒄𝒉𝒐𝒓𝒅). [With the given image]

How to prove that: The line drawn from the centre of a circle to the midpoint of a chord is perpendicular to the chord. (𝒍𝒊𝒏𝒆 𝒇𝒓𝒐𝒎 𝒄𝒆𝒏𝒕𝒓𝒆 𝒕𝒐 𝒎𝒊𝒅𝒑𝒕 𝒐𝒇 𝒄𝒉𝒐𝒓𝒅) [With the given image]

How to prove that: The angle subtended by an arc at the centre of a circle is double the size of the angle subtended by the same arc at the circumference (on the same side of the chord as the centre). (∠ 𝒂𝒕 𝒄𝒆𝒏𝒕𝒓𝒆 = 𝟐 × ∠ 𝒂𝒕 𝒄𝒊𝒓𝒄𝒖𝒎𝒇𝒆𝒓𝒆𝒏𝒄𝒆) [With the given image]

(When drawing remember to label each angle the extension creates)

How to prove that: The opposite angles of cyclic quadrilateral are supplementary. (𝒐𝒑𝒑 ∠ ′𝒔 𝒐𝒇 𝒄𝒚𝒄𝒍𝒊𝒄 𝒒𝒖𝒂𝒅) [With the given image]

How to prove that: The angle between the tangent to a circle and a chord drawn from the point of contact are equal to the angle in the alternate segment. (𝒕𝒂𝒏 𝒄𝒉𝒐𝒓𝒅 𝒕𝒉𝒆𝒐𝒓𝒆𝒎) [With the given image - prove 𝐵𝑃̂𝑇 = 𝐴̂)

How to prove: Proportional division theorem [With the given image]

CONSTRUCTION:

1. Draw altitudes h (base AD) and K (base AE)

2. Join BE and DC to create ΔBDE and ΔCED

PROOF:

1. Area ΔADE / Area ΔBDE = (¹/₂ . AD . h) / (¹/₂ . BD . h) = AD / BD

2. Area ΔADE / Area ΔCDE = (¹/₂ . AE . h) / (¹/₂ . CE . h) = AE / CE

3. Area ΔBDE = Area ΔCDE (Same base, same height, lying between ∥ lines)

4. Area ΔADE / Area ΔBDE = Area ΔADE / Area ΔCDE

5. AD / BD = AE / BC

How to prove that: 2 triangles are similar [With the given image]

CONSTRUCTION:

1. Label P, where AP = DE

2. Label Q, where AQ = DF

3. Join P and Q to form PQ

PROOF:

1. In ΔAPQ and ΔDEF
   1. ∠A = ∠D Given
   2. AP = DE Construction
   3. AQ = DF Construction
   ΔAPQ ≡ ΔDEF (s,∠,s)

2. ∠APQ = ∠E and ∠AQP = ∠F (ΔAPQ ≡ ΔDEF)
   ∠APQ = ∠B = ∠E (Given)
   PQ ∥ BC (Corresponding ∠s are =)

3. AP / AB = AQ / AC (ΔABC; PQ ∥ BC)
   But AP = DE and AQ = DF

4. DE / AB = DF / AC

What is the sum of interior angles of a polygon with n sides?

(n−2) × 180°

What is the size of each interior angle of a regular polygon with n sides?

[(n−2) × 180°] ÷ n

What is the circumference of a circle with radius r?

C = 2πr

What is the circumference of a circle with diameter d?

C = πd

What is the area of a circle with radius r?

A = πr²

What is the surface area of a sphere with radius r?

SA = 4πr²

What is the volume of a sphere with radius r?

V = (4/3)πr³

What is the curved surface area of a hemisphere with radius r?

Curved SA = 2πr²

What is the total surface area of a hemisphere with radius r?

Total SA = 3πr²

What is the volume of a hemisphere with radius r?

V = (2/3)πr³

What is the volume of a cylinder with radius r and height h?

V = πr²h

What is the curved surface area of a cylinder with radius r and height h?

Curved SA = 2πrh

What is the total surface area of a cylinder with radius r and height h?

Total SA = 2πr² + 2πrh

What is the volume of a cone with radius r and height h?

V = (1/3)πr²h

What is the curved surface area of a cone with radius r and slant height l?

Curved SA = πrl

What is the total surface area of a cone with radius r and slant height l?

Total SA = πr² + πrl

What is the slant height of a cone with radius r and perpendicular height h?

l = √(r² + h²)

What is the volume of a triangular prism with base b, triangle height h, and length l?

V = (1/2)bhl

What is the volume of a square prism with side s and height h?

V = s²h

What is the total surface area of a square prism with side s and height h?

SA = 2s² + 4sh

What is the volume of a rectangular prism with length l, width w, and height h?

V = lwh

What is the total surface area of a rectangular prism with length l, width w, and height h?

SA = 2(lw + lh + wh)

What is the volume of a square-based pyramid with base side s and height h?

V = (1/3)s²h

What is the total surface area of a square-based pyramid with base side s and slant height l?

SA = s² + 2sl

What is the area of a triangle with base b and perpendicular height h?

A = (1/2)bh

What is the area of a triangle with two sides a and b and included angle C?

A = (1/2)ab sinC

What is the area of a kite with diagonals d₁ and d₂?

A = (1/2)d₁d₂

What is the area of a rhombus with diagonals d₁ and d₂?

A = (1/2)d₁d₂

What is the area of a rhombus with base b and perpendicular height h?

A = bh

What is the perimeter of a rhombus with side length s?

P = 4s

What is the area of a parallelogram with base b and perpendicular height h?

A = bh

What is the perimeter of a parallelogram with sides a and b?

P = 2(a + b)

What is the area of a rectangle with length l and width w?

A = lw

What is the perimeter of a rectangle with length l and width w?

P = 2(l + w)

What is the diagonal of a rectangle with length l and width w?

d = √(l² + w²)

What is the area of a trapezium with parallel sides a and b and perpendicular height h?

A = (1/2)(a + b)h

What is the area of a square with side s?

A = s²

What is the perimeter of a square with side s?

P = 4s

What is the diagonal of a square with side s?

d = s√2

If a shape is enlarged by linear scale factor k, by what factor does its area change?

k²

If a shape is enlarged by linear scale factor k, by what factor does its volume change?

k³

What is the distance between two points (x₁, y₁) and (x₂, y₂)?

d = √[(x₂−x₁)² + (y₂−y₁)²]

What is the midpoint of a line segment between (x₁, y₁) and (x₂, y₂)?

M = ((x₁+x₂)/2 ; (y₁+y₂)/2)

What is the gradient of a line through (x₁, y₁) and (x₂, y₂)?

m = (y₂−y₁) ÷ (x₂−x₁)

What is the equation of a circle with centre (a, b) and radius r?

(x−a)² + (y−b)² = r²

What is the length of an arc with radius r and angle θ (in degrees)?

Arc = (θ/360) × 2πr

What is the area of a sector with radius r and angle θ (in degrees)?

Sector area = (θ/360) × πr²

[LINES] The adjacent angles on a straight line are supplementary.

∠s on a str line

[LINES] If the adjacent angles are supplementary, the outer arms of these angles form a straight line.

adj ∠s supp

[LINES] The adjacent angles in a revolution add up to 360°.

∠s round a pt OR ∠s in a rev

[LINES] Vertically opposite angles are equal.

vert opp ∠s =

[LINES] If AB || CD, then the alternate angles are equal.

alt ∠s; AB || CD

[LINES] If AB || CD, then the corresponding angles are equal.

corresp ∠s; AB || CD

[LINES] If AB || CD, then the co-interior angles are supplementary.

co-int ∠s; AB || CD

[LINES] If the alternate angles between two lines are equal, then the lines are parallel.

alt ∠s =

[LINES] If the corresponding angles between two lines are equal, then the lines are parallel.

corresp ∠s =

[LINES] If the co-interior angles between two lines are supplementary, then the lines are parallel.

coint ∠s supp

[TRIANGLES] The interior angles of a triangle are supplementary.

∠ sum in Δ OR sum of ∠s in Δ OR Int ∠s Δ

[TRIANGLES] The exterior angle of a triangle is equal to the sum of the interior opposite angles.

ext ∠ of Δ

[TRIANGLES] The angles opposite the equal sides in an isosceles triangle are equal. (i.e. Given image - Prove B and C equal)

∠s opp equal sides

[TRIANGLES] The sides opposite the equal angles in an isosceles triangle are equal. (i.e. Given image - prove c and b equal)

sides opp equal ∠s

[TRIANGLES] In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Pythagoras OR Theorem of Pythagoras

[TRIANGLES] If the square of the longest side in a triangle is equal to the sum of the squares of the other two sides then the triangle is right-angled.

Converse Pythagoras OR Converse Theorem of Pythagoras

[TRIANGLES] If three sides of one triangle are respectively equal to three sides of another triangle, the triangles are congruent.

SSS

[TRIANGLES] If two sides and an included angle of one triangle are respectively equal to two sides and an included angle of another triangle, the triangles are congruent.

SAS OR S∠S

[TRIANGLES] If two angles and one side of one triangle are respectively equal to two angles and the corresponding side in another triangle, the triangles are congruent.

AAS OR ∠∠S

[TRIANGLES] If in two right-angled triangles, the hypotenuse and one side of one triangle are respectively equal to the hypotenuse and one side of the other, the triangles are congruent.

RHS OR 90°HS

[TRIANGLES] The line segment joining the midpoints of two sides of a triangle is parallel to the third side and equal to half the length of the third side. (i.e. Given image - prove DE parallel to BC AND BC = 2DE)

Midpt Theorem

[TRIANGLES] The line drawn from the midpoint of one side of a triangle, parallel to another side, bisects the third side. (i.e. Given image - Given AD = AB and DE is parallel to BC, prove AE = EC)

line through midpt || to 2nd side

[TRIANGLES] A line drawn parallel to one side of a triangle divides the other two sides proportionally. (i.e. Given image - Given that HF is parallel to BC, Prove AH/HB = AF/FC

line || one side of Δ OR prop theorem; name || lines (i.e. - In ΔABC, HF || BC)

[TRIANGLES] If a line divides two sides of a triangle in the same proportion, then the line is parallel to the third side. (i.e. Given image - Prove that HF is parallel to BC)

line divides two sides of Δ in prop

[TRIANGLES] If two triangles are equiangular, then the corresponding sides are in proportion (and consequently the triangles are similar).

||| Δs OR equiangular Δs

[TRIANGLES] If the corresponding sides of two triangles are proportional, then the triangles are equiangular (and consequently the triangles are similar).

Sides of Δ in prop

[TRIANGLES] If triangles (or parallelograms) are on the same base (or on bases of equal length) and between the same parallel lines, then the triangles (or parallelograms) have equal areas.

same base; same height OR equal bases; equal height

[CIRCLES] The tangent to a circle is perpendicular to the radius/diameter of the circle at the point of contact.

tan ⊥ radius OR tan ⊥ diameter

[CIRCLES] If a line is drawn perpendicular to a radius/diameter at the point where the radius/diameter meets the circle, then the line is a tangent to the circle.

line ⊥ radius OR converse tan ⊥ radius OR converse tan ⊥ diameter

[CIRCLES] The line drawn from the centre of a circle to the midpoint of a chord is perpendicular to the chord. (i.e. Given Image - Prove Angle ACE is perpendicular)

line from centre to midpt of chord

[CIRCLES] The line drawn from the centre of a circle perpendicular to a chord bisects the chord. (i.e. Given image - Prove that DC = EC)

line from centre ⊥ to chord

[CIRCLES] The perpendicular bisector of a chord passes through the centre of the circle.

perp bisector of chord

[CIRCLES] The angle subtended by an arc at the centre of a circle is double the size of the angle subtended by the same arc at the circle (on the same side of the chord as the centre).

∠ at centre = 2 × ∠ at circumference

[CIRCLES] The angle subtended by the diameter at the circumference of the circle is 90°.

∠s in semi-circle OR diameter subtends right angle OR ∠ in ½

[CIRCLES] If the angle subtended by a chord at the circumference of the circle is 90°, then the chord is a diameter.

chord subtends 90° OR converse ∠s in semi-circle

[CIRCLES] Angles subtended by a chord of the circle, on the same side of the chord, are equal.

∠s in the same seg

[CIRCLES] If a line segment joining two points subtends equal angles at two points on the same side of the line segment, then the four points are concyclic.

line subtends equal ∠s OR converse ∠s in the same seg

[CIRCLES] Equal chords subtend equal angles at the circumference of the circle.

equal chords; equal ∠s (Angle A = Angle D)

[CIRCLES] Equal chords subtend equal angles at the centre of the circle. (i.e. Given image, prove that angle FAC is = angle EAD)

equal chords; equal ∠s

[CIRCLES] Equal chords in equal circles subtend equal angles at the circumference of the circles.

equal circles; equal chords; equal ∠s

[CIRCLES] Equal chords in equal circles subtend equal angles at the centre of the circles.

equal circles; equal chords; equal ∠s

[CIRCLES] The opposite angles of a cyclic quadrilateral are supplementary.

opp ∠s of cyclic quad

[CIRCLES] If the opposite angles of a quadrilateral are supplementary then the quadrilateral is cyclic.

opp ∠s quad supp OR converse opp ∠s of cyclic quad

[CIRCLES] The exterior angle of a cyclic quadrilateral is equal to the interior opposite angle.

ext ∠ of cyclic quad

[CIRCLES] If the exterior angle of a quadrilateral is equal to the interior opposite angle of the quadrilateral, then the quadrilateral is cyclic.

ext ∠ = int opp ∠ OR converse ext ∠ of cyclic quad

[CIRCLES] Two tangents drawn to a circle from the same point outside the circle are equal in length.

Tans from common pt OR Tans from same pt