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Date May 2017 Marks available 3 Reference code 17M.1.hl.TZ0.12
Level HL only Paper 1 Time zone TZ0
Command term Show that Question number 12 Adapted from N/A

Question

The vertices A, B, C of an acute angled triangle have position vectors a, b, c with respect to an origin O.

The mid-point of [BC] is denoted by D. The point E lies on [AD] such that \({\text{AE}} = 2{\text{DE}}\).

The perpendiculars from B to [AC] and C to [AB] meet at the point F.

Show that the position vector of E is

\(\frac{1}{3}\) (abc).

[4]
a.i.

Explain briefly why this result shows that the three medians of a triangle are concurrent.

[1]
a.ii.

Show that the position vector f of F satisfies the equations

(b f ) \( \bullet \) (c a) = 0

(c f ) \( \bullet \) (a b) = 0.

[3]
b.i.

Show, by expanding these equations, that

(af ) \( \bullet \) (cb) = 0.

[3]
b.ii.

Explain briefly why this result shows that the three altitudes of a triangle are concurrent.

[1]
b.iii.

Markscheme

M17/5/FURMA/HP1/ENG/TZ0/12.a.i/M

d = b + \(\frac{1}{2}\)(cb) = \(\frac{1}{2}\)(b + c)     (M1)A1

e = d + \(\frac{1}{3}\)(ad)     M1

= \(\frac{1}{2}\)(b + c) + \(\frac{1}{3}\)(a - \(\frac{1}{2}\)(b + c))     A1

= \(\frac{1}{3}\)(a + b + c)     AG

[??? marks]

a.i.

(because of the symmetry of the result), the other two medians also pass through E.     R1

[??? marks]

a.ii.

M17/5/FURMA/HP1/ENG/TZ0/12.b.i/M

\(\overrightarrow {{\text{BF}}} \) = fb and \(\overrightarrow {{\text{AC}}} \) = ca     A1

since FB is perpendicular to AC, (bf) \( \bullet \) (ca) = 0     R1AG

similarly since FC is perpendicular to BA, (cf) \( \bullet \) (ab) = 0     R1AG

[??? marks]

b.i.

expanding these equations and adding,     M1

b \( \bullet \) cb \( \bullet \) af \( \bullet \) c + f \( \bullet \) a + c \( \bullet \) ac \( \bullet \) bf \( \bullet \) a + f \( \bullet \) b = 0     A1

b \( \bullet \) af \( \bullet \) c + c \( \bullet \) a + f \( \bullet \) b = 0     A1

leading to (af) \( \bullet \) (cb) = 0     AG

[??? marks]

b.ii.

this result shows that AF is perpendicular to BC so that the three altitudes are concurrent (at F)     R1

[??? marks]

b.iii.

Examiners report

[N/A]
a.i.
[N/A]
a.ii.
[N/A]
b.i.
[N/A]
b.ii.
[N/A]
b.iii.

Syllabus sections

Topic 2 - Geometry » 2.2

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