Cartesian plane
Question
Two vertices of a triangle are (4,-3) and (-2,5). If the orthocenter of the triangle is at (1,2), then the third vertex is
Moderate
Solution
Let the third vertex be (h,k)
Now, the slope of AD is (k−2)/(h−1), the slope of BC is (5+3)/(−2−4)=−4/3,the slope of BE is (−3−2)/(4−1)=−5/3, and the slope of AC is (k−5)/(h+2). Since AD⊥BC, we have
k−2h−1×−43=−1
or 3h−5k+31=0_______(i)
Again, since BE⊥AC,, we have
−53×k−5h+2=−1
or 3h−5k+31=0______(ii)
On solving (i) and (ii), we get h=33, k=26. Hence, the third vertex is(33,26)
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