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May 14, 2019

Two Similar Triangles, ISI Entrance Subjective 2018

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Understand the problem

[/et_pb_text][et_pb_text _builder_version="3.27.4" text_font="Raleway||||||||" background_color="#f4f4f4" custom_margin="10px||10px" custom_padding="10px|20px|10px|20px" box_shadow_style="preset2"]Suppose that \(PQ\) and \(RS\) are two chords of a circle intersecting at a point \(O\) , It is given that \(PO=3\) cm and \( SO=4\) cm . Moreover, the area of the triangle \(POR\) is \(7 cm^2 \) . Find the are of the triangle \(QOS\) .  

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I.S.I. (Indian Statistical Institute) B.Stat/B.Math Entrance Examination 2018. Subjective Problem no. 2.
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Plane Geometry [/et_pb_accordion_item][et_pb_accordion_item title="Difficulty Level" _builder_version="3.22.4" open="off"]

5.5 out of 10

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'Challenge and Thrill of Pre-College Mathematics' by V,Krishnamurthy, C.R.Pranesachar, ect.

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Start with hints

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Do you really need a hint? Try it first!

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\(PQ\) and \( RS\) are two chords of the circle \(C\) , intersecting at the point \(O\). See figure: click here.

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Given \(PO=3\) cm \(SO=4\) cm \([\triangle POR]= 7 cm^2\).  

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From the triangles \(POS\) and \(QOS\) we have,                 \(\angle POR=\angle SOQ\) [Opposite angles]                 \(\angle SRP=\angle SQO \) [Angle on the same semi-circle \(STP\)]                 \(\angle QSO= \angle OPR\) [Angle on the same semi-circle \(ST'P\)]  Therefore the \(\triangle POR\) and \(\triangle SOQ\) are similar triangles .  

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\(\frac{[\triangle POR]}{OP^2}=\frac{[\triangle SOQ]}{SO^2}.\) \(\Rightarrow [\triangle SOQ]=\frac{SO^2}{PO^2}\cdot [\triangle POR]\)=\(\frac{4^2}{3^2}\cdot 7=12\frac{4}{9}\).(Ans.)

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Similar Problem

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One comment on “Two Similar Triangles, ISI Entrance Subjective 2018”

  1. Area of △POR/Area of △QOS=(PO/SO)^2; putting known values;
    or 7 cm 2/Area of △QOS=(3 cm/ 4 cm )^2=9/16;
    or (7 cm ^2) *(16/9)=Area of △QOS;
    Therefore Area of △QOS= (112/9) cm^2=12.4(recurring) cm ^2 answer

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