Grade 12 Maths · Euclidean Geometry · Prove a line is parallel to a side

Prove a line is parallel to a side Grade 12

Prove that a line is parallel to one side of a triangle by showing it divides the other two sides in the same ratio, for example AD/DB = AE/EC = 2/3 gives DE ∥ BC.

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Practice · NavyBlue · 2 marks

Δ K L M

P

lies on

K L

and

Q

lies on

K M

K P = x + 1

P L = 2 x

K Q = 4

and

QM = 6

PQ

is not parallel to

L M

when

x = 2

.(2)

Work it out on paper first. Then mark yourself with the memo, one tick at a time.

More questions: prove a line is parallel to a side

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3. Practice · NavyBlue · 4 marks
In $Δ PQR$ , $S$ lies on $PQ$ and $T$ lies on $PR$ . $PS = 8$ , $SQ = 12$ , $PT = 10$ and $PR = 25$ .
(b) Prove that $ST ∥ QR$ .(4)
Show the memo (4 ticks)Memo
1. ✓
  Ratio on PQ $\frac{PS}{SQ} = \frac{8}{12} = \frac{2}{3}$
  1 mark · Prove a line is parallel to a side
2. ✓
  Ratio on PR $\frac{PT}{TR} = \frac{10}{15} = \frac{2}{3}$
  1 mark · Prove a line is parallel to a side
3. ✓
  Statement and reason $\frac{PS}{SQ} = \frac{PT}{TR}$ , so $ST ∥ QR$ [line divides two sides of $Δ$ in prop]
  2 marks · Prove a line is parallel to a side
4. Practice · NavyBlue · 4 marks
In $Δ A BC$ , $D$ lies on $A B$ and $E$ lies on $A C$ . $A D = 6$ , $D B = 4$ , $A E = 9$ and $EC = 6$ .
(a) Prove that $D E ∥ BC$ .(4)
Show the memo (4 ticks)Memo
1. ✓
  Ratio on side AB $\frac{A D}{D B} = \frac{6}{4} = \frac{3}{2}$
  1 mark · Prove a line is parallel to a side
2. ✓
  Ratio on side AC $\frac{A E}{EC} = \frac{9}{6} = \frac{3}{2}$
  1 mark · Prove a line is parallel to a side
3. ✓
  Statement and reason $\frac{A D}{D B} = \frac{A E}{EC}$ , so $D E ∥ BC$ [line divides two sides of $Δ$ in prop]
  2 marks · Prove a line is parallel to a side
5. Practice · NavyBlue · 4 marks
In $Δ A BC$ , $D$ lies on $A B$ , $E$ lies on $A C$ and $F$ lies on $BC$ . $D E ∥ BC$ , $A E = 4$ , $EC = 6$ , $BF = 9$ and $FC = 6$ .
(b) Prove that $D F ∥ A C$ .(4)
Show the memo (4 ticks)Memo
1. ✓
  Ratio on BA $\frac{B D}{D A} = \frac{3}{2}$
  1 mark · Prove a line is parallel to a side
2. ✓
  Ratio on BC $\frac{BF}{FC} = \frac{9}{6} = \frac{3}{2}$
  1 mark · Prove a line is parallel to a side
3. ✓
  Statement and reason $\frac{B D}{D A} = \frac{BF}{FC}$ , so $D F ∥ A C$ [line divides two sides of $Δ$ in prop]
  2 marks · Prove a line is parallel to a side

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