Hi,
Thanks. But I'm rather confused.
How are we certain that drawing 2 vertical lines make the rectangles into 3 equal parts?

From the pictures, I divide it to 3 shaded parts, A,B and C.
Shaded part A = Unshaded part A
Shaded part B = Unshaded part B
Shaded part C = Unshaded part C.
Therefore, shaded part = unshaded part.
Meaning shaded part = ½ of total figure.
While unshaded part also another half of total figure.

Another way to think about it : (since you have learnt area of triangle in P5)

The two shaded triangles WCD and CXY both have a height of 3cm. Their combined base is 12cm

Area of the two triangles

= ½ x 3cm x base of WCD + ½ x 3cm x base of CXY

= ½ x 3cm x (combined base)
= ½ x 3cm x 12cm
= ½ x 36cm²
= 18cm²

We don't know what length WC and CX is.

Let's say WC = 9cm and CX = 3cm

Area of two triangles

= ½ x 3cm x 9cm + ½ x 3 x 3cm
= ½ x 3cm x (9cm + 3cm)
= ½ x 3cm x 12cm
= 18cm²

Now ½ x 3 = 3/2 or 1.5 or 1½

We can see it as adding 9 groups of 1½ to 3 groups of 1½, to get 12 groups of 1½.


The above is an hypothetical example to show the distributive law of mathematics.

So note that we cannot actually assume that WC = 9cm, CX = 3cm since we only know the sum is 12cm.


It could have been (8cm,4cm),(7cm,5cm) (8.88cm,3.12cm) among other combinations whereby ① sum of WC and CX is 12cm and ② WC is longer than CX (based on the diagram)

The answer would still be 18cm² as long as ① is fulfilled.



So generally :

WC + CX = 12cm

Area of these two triangles

= ½ x 3cm x WC + ½ x 3cm x CX
= ½ x 3 x (WC + CX)
= ½ x 3cm x 12cm
= 18cm²

Thanks for the detailed explanation. Now it makes sense.

Welcome.


To answer your updated question, the rectangle isn't necessarily divided into 3 smaller identical rectangles of equal area.

Because there is no length given that confirms this.

Rather, Mr AC Lim has divided it into 3 rectangles, and each of these rectangles has been divided into two equal parts.

We know that the diagonals of a rectangle or square always divides it equally into two.

So what we see is :

2 triangle A (one shaded, one unshaded,
both are identical except their colour. They have equal areas)

2 triangle B (same logic as above)
2 triangle C (same logic as above)


We can rearrange them into two groups :

①A B C (all shaded)
②A B C (all unshaded)

Now the shaded part is just made of shaded A, shaded B, shaded C.

Their total area is equal to unshaded A + unshaded B + unshaded C

Basically, out of the 2 groups of triangles only 1 group is shaded.

So that's ½ the area of the whole figure (The whole figure is a rectangle)

Therefore, area of shaded part
= ½ the area of rectangle
= ½ x 3cm x 12cm
= 18cm²

The division of a rectangle or square into two equal parts by drawing the diagonal, is the very basis/fundamental to explain why the area of triangle has a formula ½ x base x height.

It is most intuitive/obvious for right-angled triangles.

For non right-angle triangles (viewing obtuse triangles and acute triangles in some perspectives), there is some further explanation but the formula still holds.

Hi J,
Thanks for the extended explanation.
It was not an updated question.
Just now, my question disappeared. I just re-submit it in case others don't know why you are explaining.

Sorry and truly appreciated.

Wow ...Very well explained @J.

Oh no worries. Because I saw the first comment and it was different from earlier in the day.

Hope my explanation was of help and you have gained a better understanding of the question and concept.

Definitely! ; )
Thanks