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Question

If you are asked to construct APQ ABC with the scale factor 35, which of the following are correct?  In ABC,AB=4cm, BC=3cm and ABC=90.​


 
  1. AQP=ACB
  2. AA3P=AA5P
  3. AP:PB=3:2
  4. 5PQ = 3BC


Solution

The correct options are
A AQP=ACB
C AP:PB=3:2
D 5PQ = 3BC
The following steps will give you the information on how to construct the similar triangle to ABC.
Step 1: Draw a line AB = 4 cms.
Step2: Draw a line BC = 3 cm, perpendicular to AB passing through B.
Step 3: Join AC.
Step 4: Draw a ray AX, making an acute angle with line AB.
Step 5: Mark 5 points A1,A2,A3,A4 and A5 such that A1A2=A2A3=A3A4=A4A5.

Step 6: Join BA5.
Step 7: Draw a line parallel to BA5 passing through A3 by making an angle equal to AA5B, intersecting AB at the point P.APAB=35.(This is the given scale factor of the smaller triangle which is the ratio of corresponding sides.)
Step 8: Draw a line parallel to BC passing through P, intersecting AC at Q.
Now we have constructed the triangle APQ ABC​.
By basic proportionality theorem,
The corresponding angles are equal. Therefore in ΔABC and ΔAPQ,BAC=PAQ, ABC=APQ and ACB=PQA.
And the corresponding sides are proportional. Therefore in ΔABC and ΔAPQ,APAB=PQBC=AQAC
We know that APAB=35
Therefore, PQBC=35
5PQ = 3BC
From construction we have, A3P||A5B.AA3P=AA5B, because they are corresponding angles of parallel lines.Now AA3PAA5P as it is equal to AA5B. 
Consider triangle AA5B, the line A3P being parallel to A5B cuts the sides AB and AA5 in the same proportion.
i.e. AA3A3A5=APPB
APPB=32
Therefore A3P will divide the line AB in the ratio 3:2.
 

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