# Math problem .Knowing A(1;-2) B(3;t) C(9;-3) find t for the angle BAC=90 degrees .

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Knowing A(1;-2) B(3;t) C(9;-3) we have to find t for the angle BAC=90 degrees

The slope of the line AB is (t + 2)/(3 - 1) = (t + 2)/2

The slope of AC is (-2 + 3)/(1 - 9) = -1/8

As BAC is equal to 90 degrees AB and AC are perpendicular. So the product of the slope is -1.

[(t + 2)/2 ]*(-1/8) = -1

=> (t + 2)/2 = 8

=> (t + 2) = 16

=> t = 14

**The required value of t = 14.**

If the measure of the angle BAC = 90 degrees, then the triangle ABC is a right angle triangle and BC is the hypotenuse.

We'll apply the Pythagorean theorem in the triangle ABC:

BC^2 = AB^2 + AC^2

BC^2 = (xC - xB)^2 + (yC - yB)^2

BC^2 = (9 - 3)^2 + (-3 - t)^2

BC^2 = 36 + 9 + 6t + t^2

BC^2 = 45 + 6t + t^2 (1)

AC^2 = (xC - xA)^2 + (yC - yA)^2

AC^2 = (9-1)^2 + (-3+2)^2

AC^2 = 64 + 1

AC^2 = 65 (2)

AB^2 = (xB - xA)^2 + (yB - yA)^2

AB^2 = (3 - 1)^2 + (t+2)^2

AB^2 = 4 + t^2 + 4t + 4

AB^2 = t^2 + 4t + 8 (3)

We'll substitute (1) , (2) , (3) in the formula of Pythagorean Theorem:

BC^2 = AB^2 + AC^2

45 + 6t + t^2 = t^2 + 4t + 8 + 65

We'll combine and eliminate like terms:

45 + 6t - (4t + 8 + 65) = 0

2t - 28 = 0

2t = 28

**t = 14**