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 [tex]x=\frac{2+7}{2}=\frac{9}{2}=4.5[/tex]
∴ 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)
∵ [tex]m_{AC}=\frac{y-3}{4.5-2}=\frac{y-3}{2.5}[/tex]
∵ [tex]m_{BC}=\frac{y-3}{4.5-7}=\frac{y-3}{-2.5}[/tex]
∵ [tex]m_{AC}*m_{BC}=-1[/tex]
∴ [tex]\frac{y-3}{2.5}*\frac{y-3}{-2.5}=-1[/tex]
- 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)
Answer: c(4.5,5.5) or c(4.5,0.5)
Step-by-step explanation:
Mac=y-3/4.5-2=y-3/2.5
Mbc=y-3/4.5-7=y-3/-2.5
