Singapore Find The Area Rectangle
Inside of rectangle ABCD is a triangle AMN with M along BC and N along CD. Suppose BM = 6 cm, ND = 4 cm, and the area of AMN = 70 cm2. What is the area of the rectangle ABCD ?
This was asked to primary 6 school students in Singapore, aged about 12 to 13 years old.
As usual, watch the video for a solution.
Singapore Find The Area Rectangle
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Answer To Singapore Find The Area Rectangle
(Pretty much all posts are transcribed quickly after I make the videos for them-please let me know if there are any typos/errors and I will correct them, thanks).
This is quite a challenging problem and I did not see how to solve it at first either. The key is to break the rectangle down into four rectangles, and break the triangle down into three triangles.
Construct NP parallel to side AD, and construct MQ parallel to side AB. Let NP and MQ intersect at R. We have created 4 rectangles inside ABCD, so we have AP = ND = 4 cm, and AQ = BM = 6 cm.
Triangle ARN has a base of RN and a height equal to AP = QR = 4 cm. Triangle ARN has an area equal to 0.5(RN)AP. Rectangle DQRN has an area equal top (RN)(QR). Since AP = QR, we can conclude triangle ARN has 1/2 the area of the rectangle DQRN.
Similarly, triangle ARM has 1/2 the area of the rectangle BMRP, and triangle NRM has 1/2 the area of the rectangle CNRM. So we have:
areas
ARN = 0.5(DQRN)
ARM = 0.5(BMRP)
NRM = 0.5(CNRM)
The sum of the areas on the left hand side will be that of the triangle AMN = 70 cm2. This is equal to the sum of the areas on the right hand side of the three equations. So we have:
area
0.5(DQRN)+ 0.5(BMRP) + 0.5(CNRM) = 70
DQRN+ BMRP + CNRM = 140
Now add the area of APRQ to both sides of the equation:
area
DQRN+ BMRP + CNRM + APRQ = 140 + APRQ
But we know APRQ has sides of 4 cm and 6 cm, so its area is 4×6 = 24 cm2. Thus we have:
DQRN + BMRP + CNRM + APRQ = 140 + 24
DQRN + BMRP + CNRM + APRQ = 164
The sum of the four rectangles is also precisely the entire area of ABCD. So we have the answer. The area of ABCD is 164 cm2.
ABCD = 164 cm2
Reference
Facebook group
https://www.facebook.com/groups/259454320877898/permalink/1247815375375116/
https://mindyourdecisions.com/blog/2025/01/15/singapore-find-the-area-rectangle/
https://mindyourdecisions.com/blog/?p=37553