Question

How to identify two points P and Q on a line segment AB such that AP:PQ = 1:2 and PQ:QB = 4:5?

A

Draw a ray at an acute angle to AB and identify the points P and Q on AB with ratio 1:2 and 4:5 respectively.

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B

Find the ratios in which the points P and Q divide line segment AB and then divide the line segment AB in the ratios obtained.

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C

The points P and Q are points of trisection of the line segment AB. Identify the points P and Q by dividing AB into three equal parts.

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D

None of the above

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Solution

The correct option is A Find the ratios in which the points P and Q divide line segment AB and then divide the line segment AB in the ratios obtained. Given AP:PQ = 1:2 and PQ:QB = 4:5. APPQ=12⇒APPQ+1=12+1⇒AP+PQPQ=32⇒AQPQ=32−−−−−−−−−−−−−(1)QBPQ=54−−−−−−−−−−−−−−−(2) Dividing equation (1) by (2) ⇒AQQB=65−−−−−−−−−−−−(3) Hence Q divides AB in the ratio of 6:5. Adding equations (1) and (2), ⇒AQPQ+QBPQ=32+54⇒ABPQ=114−−−−−−(4) Dividing APPQ=12 with equation (4), we get APAB=211 Hence P divides line segment AB in the ratio of 2:11. Now proceed further by dividing line segment AB by the two ratios respectively and thereby identifying the points.

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