S2 Mathematics Practice Book · Unit 5: Thales theorems

Unit 5 Thales theorems

Key unit competence: Apply Thales' theorem in solving problems related to similar triangles and trapezia.

Key ideas

Midpoint theorem: the segment joining the midpoints of two sides of a triangle is parallel to the third side and half its length.
Thales' theorem (triangle): if a line parallel to one side of a triangle cuts the other two sides, it divides them proportionally. In $△ A BC$ with $D \in A B$ , $E \in A C$ and $D E ∥ BC$ : $\frac{A D}{D B} = \frac{A E}{EC}$ and $\frac{A D}{A B} = \frac{A E}{A C} = \frac{D E}{BC}$ .
Thales in a trapezium: a line parallel to the parallel sides of a trapezium divides the non-parallel sides proportionally.
Converse: if a line divides two sides of a triangle in the same ratio, it is parallel to the third side. If the ratios are different, the line is not parallel.
To solve length problems: identify the parallel lines, write ratios starting from the same vertex, cross-multiply and solve.

Formulas to know

\frac{A D}{D B} = \frac{A E}{EC}

\frac{A D}{A B} = \frac{A E}{A C} = \frac{D E}{BC}

MN = \frac{1}{2} BC

Common misconceptions

" $\frac{A D}{D B} = \frac{D E}{BC}$ ." No: $\frac{D E}{BC}$ compares with the whole sides: $\frac{A D}{A B} = \frac{D E}{BC}$ .
"Any line across a triangle divides the sides proportionally." Only a line parallel to the third side does.

Worked examples

Example 1. In $△ A BC$ , $D E ∥ BC$ with $D$ on $A B$ , $E$ on $A C$ . $A D = 4$ , $D B = 6$ , $A E = 5$ . Find $EC$ .

Solution

$\frac{A D}{D B} = \frac{A E}{EC} \Rightarrow \frac{4}{6} = \frac{5}{EC} \Rightarrow EC = 7.5$ .

Example 2. $M$ and $N$ are midpoints of $A B$ and $A C$ , and $BC = 14$ cm. Find $MN$ .

Solution

By the midpoint theorem, $MN ∥ BC$ and $MN = \frac{1}{2} \times 14 = 7$ cm.

Practice questions

Easier and harder questions are mixed, just like in a real exam.

State the midpoint theorem and draw a diagram.
Show answer
In $△ A BC$ , if $M$ and $N$ are the midpoints of $A B$ and $A C$ , then $MN ∥ BC$ and $MN = \frac{1}{2} BC$ .
Common mistake: Saying $MN = BC$ .
$D E ∥ BC$ , $A D = x - 1$ , $D B = x + 3$ , $A E = 2$ , $EC = 6$ . Find $x$ .
Show answer
$\frac{x - 1}{x + 3} = \frac{2}{6}$ : $6 x - 6 = 2 x + 6$ , $x = 3$ .
Common mistake: Cross-multiplying incorrectly.
In $△ A BC$ , $D E ∥ BC$ , $A D = 6$ cm, $D B = 4$ cm and $A C = 15$ cm. Find $A E$ and $EC$ .
Show answer
$\frac{A E}{A C} = \frac{A D}{A B} = \frac{6}{10}$ , so $A E = 9$ cm and $EC = 6$ cm.
Common mistake: Using $\frac{A D}{D B}$ with the whole side $A C$ .
The diagonals of a quadrilateral join the midpoints of its sides. Explain why the quadrilateral formed by joining the midpoints of the sides of any quadrilateral is a parallelogram.
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Draw diagonal $A C$ . In triangles $A BC$ and $A D C$ , the midpoint segments are each parallel to $A C$ and equal to $\frac{1}{2} A C$ . So two opposite sides are parallel and equal: a parallelogram.
Common mistake: Assuming the original quadrilateral is a rectangle.
In $△ A BC$ , $D E ∥ BC$ , $A D = 3$ , $A B = 12$ and $BC = 16$ . Find $D E$ .
Show answer
$\frac{D E}{16} = \frac{3}{12}$ , $D E = 4$ .
Common mistake: Using $D B = 9$ instead of $A B = 12$ .
A vertical stick 1.5 m high casts a 2 m shadow. A building casts a 24 m shadow at the same time. Find the height of the building.
Show answer
$\frac{h}{24} = \frac{1.5}{2}$ , $h = 18$ m.
Common mistake: Setting up $\frac{h}{1.5} = \frac{2}{24}$ .
$A D = 4$ , $D B = 6$ , $A E = 6$ , $EC = 9$ . Show that $D E ∥ BC$ .
Show answer
$\frac{A D}{D B} = \frac{4}{6} = \frac{2}{3}$ and $\frac{A E}{EC} = \frac{6}{9} = \frac{2}{3}$ . Equal ratios, so $D E ∥ BC$ (converse of Thales).
Common mistake: Comparing $A D$ with $A E$ instead of the ratios.
In trapezium $A BC D$ , $A B ∥ EF ∥ D C$ , $E$ on $A D$ , $F$ on $BC$ , $A E = 4$ , $E D = 10$ , $BC = 21$ . Find $BF$ and $FC$ .
Show answer
$\frac{BF}{BC} = \frac{4}{14}$ , $BF = 6$ , $FC = 15$ .
Common mistake: Forgetting that $A D = A E + E D = 14$ .

Matching and drag-and-drop

Matching

Match each item in column A with the correct answer in column B. Write the letter next to each number.

Column A	Column B
1. In $△ A BC$ , $D E ∥ BC$ , $A D = 2$ , $A B = 6$ , $BC = 9$ . Find $D E$ .	a. 3
2. In $△ A BC$ , $D E ∥ BC$ , $A D = 3$ , $D B = 6$ , $A E = 4$ . Find $EC$ .	b. 15 cm
3. $\frac{A D}{D B} = \frac{A E}{EC} = \frac{2}{3}$ . What can you conclude about $D E$ and $BC$ ?	c. 8
4. A triangle has sides 8 cm, 10 cm and 12 cm. Find the perimeter of the triangle joining the midpoints of its sides.	d. $D E ∥ BC$

Show answer

1 → a (3) · 2 → c (8) · 3 → d ( $D E ∥ BC$ ) · 4 → b (15 cm)

$\frac{D E}{BC} = \frac{A D}{A B} = \frac{1}{3}$ .
Step 1. $\frac{3}{6} = \frac{4}{EC}$ .
Step 2. $EC = 8$ .
Converse of Thales' theorem.
Step 1. Midpoint theorem: each side of the new triangle is half a side of the old one: 4, 5, 6.
Step 2. Perimeter $= 4 + 5 + 6 = 15$ cm.

Drag and drop
Choose the correct word or phrase from the box for each gap. One word in the box is not needed.
$N$ is the midpoint of $A C$ · 9 · the midpoint of $A C$ · $A N = 2 NC$
1. In $△ A BC$ , $D E ∥ BC$ , $A D = 5$ , $D B = 5$ . Then $E$ is .
2. $M$ is the midpoint of $A B$ and $MN ∥ BC$ meets $A C$ at $N$ . Then .
3. $D E ∥ BC$ , $A E = 6$ , $A C = 15$ . Find $EC$ . →
Show answer
1: the midpoint of $A C$ · 2: $N$ is the midpoint of $A C$ · 3: 9
- $\frac{A E}{EC} = \frac{5}{5} = 1$ .
- A line through a midpoint parallel to another side bisects the third side.
- $EC = A C - A E = 15 - 6 = 9$ .

Matching

Match each item in column A with the correct answer in column B. Write the letter next to each number.

Column A	Column B
1. $M$ and $N$ are midpoints of sides $A B$ and $A C$ of a triangle with $BC = 10$ cm. Find $MN$ .	a. 9
2. $D E ∥ BC$ , $A D = x + 1$ , $D B = 2 x$ , $A E = 3$ and $EC = 4$ . Find $x$ .	b. $\frac{RS}{SP} = \frac{RT}{TQ}$
3. In $△ PQR$ , $ST ∥ PQ$ with $S$ on $RP$ , $T$ on $RQ$ . Which ratio is correct?	c. 2
4. $D E ∥ BC$ , $D E = 4$ , $BC = 12$ , $A D = 3$ . Find $A B$ .	d. 5 cm

Show answer

1 → d (5 cm) · 2 → c (2) · 3 → b ( $\frac{RS}{SP} = \frac{RT}{TQ}$ ) · 4 → a (9)

Midpoint theorem: $MN = \frac{1}{2} BC$ .
Step 1. $\frac{x + 1}{2 x} = \frac{3}{4}$ , so $4 (x + 1) = 6 x$ .
Step 2. $4 x + 4 = 6 x \Rightarrow x = 2$ .
Compare parts measured from the same vertex $R$ on each side.
Step 1. $\frac{A D}{A B} = \frac{4}{12}$ .
Step 2. $A B = 9$ .

Drag and drop
Choose the correct word or phrase from the box for each gap. One word in the box is not needed.
7.5 cm · 6 · 12 · 8
1. $D E ∥ BC$ , $A D = 4$ , $A B = 10$ , $A C = 15$ . Find $A E$ . →
2. Three parallel lines cut two transversals. On the first they make segments 3 cm and 5 cm. On the second, the first segment is 4.5 cm. Find the second. →
3. In a trapezium $A BC D$ with $A B ∥ EF ∥ D C$ , $E$ on $A D$ , $F$ on $BC$ : $A E = 3$ , $E D = 6$ , $BF = 4$ . Find $FC$ . →
Show answer
1: 6 · 2: 7.5 cm · 3: 8
- Step 1. $\frac{A E}{15} = \frac{4}{10}$ .
  Step 2. $A E = 6$ .
- Step 1. Parallel lines cut transversals in the same ratio: $\frac{3}{5} = \frac{4.5}{x}$ .
  Step 2. $x = \frac{5 \times 4.5}{3} = 7.5$ cm.
- $\frac{A E}{E D} = \frac{BF}{FC}$ : $\frac{3}{6} = \frac{4}{FC}$ .

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