Question

Segment AB is the hypotenuse of the right isosceles ΔABC with A(2, 3) and B(7, 3). Find all possible coordinates of C.

Answer 1

Answer:

The possible coordinates of point C are (4.5 , 5.5) OR (4.5 , 0.5)

Step-by-step explanation:

* Lets explain how to solve the problem

- The problems it seems that difficult but if you think about the

  properties of the isosceles triangle

∵ AB is the hypotenuse of the right isosceles Δ ABC

∴ The equal sides are AC and BC

∵ A = (2 , 3) and B = (7 , 3)

- The y-coordinates of A and B are equal then, AB is a horizontal

 segment

∴ The vertical segment drawn from point C to the hypotenuse AB

  will bisect it

∴ The x-coordinate of point c equal the x-coordinate of the mid-point

  of AB

∵ The x-coordinate of the mid-point of AB is half the sum of

   x-coordinates of points A and B

∴ The x-coordinate of point C is $x=\frac{2+7}{2}=\frac{9}{2}=4.5$

∴ The x-coordinate of point C is 4.5

∴ C = (4.5 , y)

* Now lets think about the slopes of the perpendicular lines

- The product of the slopes of the perpendicular line is -1

∵ ΔABC is isosceles right triangle, where m∠C = 90°

∴ AC ⊥ BC

- Lets find the slopes of AC and BC

∵ A = (2 , 3) , B = (7 , 3) and C = (4.5 , y)

∵ $m_{AC}=\frac{y-3}{4.5-2}=\frac{y-3}{2.5}$

∵ $m_{BC}=\frac{y-3}{4.5-7}=\frac{y-3}{-2.5}$

∵ $m_{AC}*m_{BC}=-1$

∴ $\frac{y-3}{2.5}*\frac{y-3}{-2.5}=-1$

- By using cross multiplication

∴ (y - 3)² = - 2.5 × 2.5 × -1

∴ (y - 3)² = 6.25

- By taking √ for both sides

∴ y - 3 = ± 2.5

∴ y - 3 = 2.5  OR  y - 3 = -2.5

∵ y - 3 = 2.5 ⇒ add 3 to both sides

∴ y = 5.5

OR

∵ y - 3 = -2.5 ⇒ add 3 to both sides

∴ y = 0.5

∴ The y-coordinates of point C are 5.5 or 0.5

* The  possible coordinates of point C are (4.5 , 5.5) OR (4.5 , 0.5)