# If A(4, 3), B(-1, y) and C(3, 4) are the vertices of a right triangle ABC, right-angled at A, then find the value of y. - Mathematics

If A(4, 3), B(-1, y) and C(3, 4) are the vertices of a right triangle ABC, right-angled at A, then find the value of y.

#### Solution

Given that A(4, 3), B(-1, y) and C(3, 4) are the vertices of the ΔABC.
ΔABC is a right triangle at A.
Hence by applying the Pythagoras Theorem, we have,
AB2 + AC2 = BC2 ....(1)
Let us find the distances, AB, BC and CA using the
distance formula.

AB=sqrt((-1-4)^2+(y-3)^2)

BC=sqrt((3+1)^2+(4-y)^2)

CA=sqrt((3-4)^2+(4-3)^2)=sqrt2

Squaring both the sides, we have

AB^2=25+y^2+9-6y

BC^2=4+16+y^2-8y

AC^2=2

Therefore, from equation (1), we have,

25+y^2+9-6y+2=4+16+y^2-8y

36+y^2-6y=20+y^2-8y

16-6y=-8y

16=-8y+6y

-y=16/2

y=-8

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