Answer:
The possible first coordinates of point C are (-2.5,1.5)
The possible second coordinates of point C are (-9.5,1.5)
Step-by-step explanation:
we know that
Triangle ABC is a right isosceles triangle
so
Is a 45°-90°-45° triangle
AC=BC
we have
A(-6,-2), B(-6,5)
step 1
Find the length side of the hypotenuse AB
[tex]AB=5-(-2)=7\ units[/tex]
step 2
Applying the Pythagoras Theorem
Find the length side of leg AC
[tex]AB^{2}=AC^{2}+BC^{2}[/tex]
Remember that
AC=BC
substitute the given values
[tex]7^{2}=AC^{2}+AC^{2}[/tex]
[tex]49=2AC^{2}[/tex]
[tex]AC^{2}=\frac{49}{2}[/tex]
[tex]AC=\frac{7\sqrt{2}}{2}\ units[/tex]
step 3
Find the first possible coordinates of C
The point C is located at right of point A
Determine the x-coordinate of point C
The x-coordinate of point C must be equal to the x-coordinate of point A plus the horizontal distance between point A and point C
Let
ACx ------> the horizontal distance between point A and point C
The horizontal distance between point A and point C is equal to the distance AC multiplied by cos(45)
[tex]ACx=(AC)cos(45\°)[/tex]
we have
[tex]cos(45\°)=\frac{\sqrt{2}}{2}[/tex]
[tex]AC=\frac{7\sqrt{2}}{2}\ units[/tex]
substitute
[tex]ACx=(\frac{7\sqrt{2}}{2})\frac{\sqrt{2}}{2}=3.5\ units[/tex]
The x-coordinate of point C is
Cx=-6+3.5=-2.5
Determine the y-coordinate of point C
The y-coordinate of point C must be equal to the y-coordinate of point A plus the vertical distance between point A and point C
Let
ACy ------> the vertical distance between point A and point C
The vertical distance between point A and point C is equal to the distance AC multiplied by sin(45)
[tex]ACy=(AC)sin(45\°)[/tex]
we have
[tex]sin(45\°)=\frac{\sqrt{2}}{2}[/tex]
[tex]AC=\frac{7\sqrt{2}}{2}\ units[/tex]
substitute
[tex]ACy=(\frac{7\sqrt{2}}{2})\frac{\sqrt{2}}{2}=3.5\ units[/tex]
The y-coordinate of point C is
Cy=-2+3.5=1.5
therefore
The possible first coordinates of point C are (-2.5,1.5)
step 4
Find the second possible coordinate of C
The point C is located at left of point A
Determine the x-coordinate of point C
The x-coordinate of point C must be equal to the x-coordinate of point A minus the horizontal distance between point A and point C
Let
ACx ------> the horizontal distance between point A and point C
The horizontal distance between point A and point C is equal to the distance AC multiplied by cos(45)
[tex]ACx=(AC)cos(45\°)[/tex]
we have
[tex]cos(45\°)=\frac{\sqrt{2}}{2}[/tex]
[tex]AC=\frac{7\sqrt{2}}{2}\ units[/tex]
substitute
[tex]ACx=(\frac{7\sqrt{2}}{2})\frac{\sqrt{2}}{2}=3.5\ units[/tex]
The x-coordinate of point C is
Cx=-6-3.5=-9.5
Determine the y-coordinate of point C
The y-coordinate of point C must be equal to the y-coordinate of point A plus the vertical distance between point A and point C
Let
ACy ------> the vertical distance between point A and point C
The vertical distance between point A and point C is equal to the distance AC multiplied by sin(45)
[tex]ACy=(AC)sin(45\°)[/tex]
we have
[tex]sin(45\°)=\frac{\sqrt{2}}{2}[/tex]
[tex]AC=\frac{7\sqrt{2}}{2}\ units[/tex]
substitute
[tex]ACy=(\frac{7\sqrt{2}}{2})\frac{\sqrt{2}}{2}=3.5\ units[/tex]
The y-coordinate of point C is
Cy=-2+3.5=1.5
therefore
The possible second coordinates of point C are (-9.5,1.5)
see the attached figure to better understand the problem
