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Question

The coordinates of the feet of the perpendiculars from the vertices of a triangle on the opposite sides are (20,25),(8,16) and (8,9). The coordinates of a vertex of the triangle are


A

(5, 10)

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B

(50, -5)

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C

(15, 30)

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D

(10, 15)

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Solution

The correct options are
A

(5, 10)


B

(50, -5)


C

(15, 30)


We use the fact that the orthocentre O of the triangle ABC is the incentre of the pedal triangle DEF.
Let (h,k) be the coordinates of O.
Now ED=(208)2+(2516)2=15

FD=20 and EF=7
so that h=7×20+20×8+15×87+20+15=10
and k=7×25+20×16+15×97+20+15=15
thus, the coordinates of O are (10,15)
Since AC is perpendicular to OE, equation of AC is
y16=108156(x8) y2x=0 (1)
Similarly equation of AB is
y9=108159(x8) 3y+x35=0 (2)
and equation of BC is
y25=20102515(x20) y+x45=0 (3)
Solving (1) and (2) we get A(5,10)
From (2) and (3) we get B(50,5) and from (3) and (1) we get C(15,30)


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