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
![AB=5-(-2)=7\ units](https://tex.z-dn.net/?f=AB%3D5-%28-2%29%3D7%5C%20units)
step 2
Applying the Pythagoras Theorem
Find the length side of leg AC
![AB^{2}=AC^{2}+BC^{2}](https://tex.z-dn.net/?f=AB%5E%7B2%7D%3DAC%5E%7B2%7D%2BBC%5E%7B2%7D)
Remember that
AC=BC
substitute the given values
![7^{2}=AC^{2}+AC^{2}](https://tex.z-dn.net/?f=7%5E%7B2%7D%3DAC%5E%7B2%7D%2BAC%5E%7B2%7D)
![49=2AC^{2}](https://tex.z-dn.net/?f=49%3D2AC%5E%7B2%7D)
![AC^{2}=\frac{49}{2}](https://tex.z-dn.net/?f=AC%5E%7B2%7D%3D%5Cfrac%7B49%7D%7B2%7D)
![AC=\frac{7\sqrt{2}}{2}\ units](https://tex.z-dn.net/?f=AC%3D%5Cfrac%7B7%5Csqrt%7B2%7D%7D%7B2%7D%5C%20units)
step 3
<em><u>Find the first possible coordinates of C</u></em>
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)
![ACx=(AC)cos(45\°)](https://tex.z-dn.net/?f=ACx%3D%28AC%29cos%2845%5C%C2%B0%29)
we have
![cos(45\°)=\frac{\sqrt{2}}{2}](https://tex.z-dn.net/?f=cos%2845%5C%C2%B0%29%3D%5Cfrac%7B%5Csqrt%7B2%7D%7D%7B2%7D)
![AC=\frac{7\sqrt{2}}{2}\ units](https://tex.z-dn.net/?f=AC%3D%5Cfrac%7B7%5Csqrt%7B2%7D%7D%7B2%7D%5C%20units)
substitute
![ACx=(\frac{7\sqrt{2}}{2})\frac{\sqrt{2}}{2}=3.5\ units](https://tex.z-dn.net/?f=ACx%3D%28%5Cfrac%7B7%5Csqrt%7B2%7D%7D%7B2%7D%29%5Cfrac%7B%5Csqrt%7B2%7D%7D%7B2%7D%3D3.5%5C%20units)
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)
![ACy=(AC)sin(45\°)](https://tex.z-dn.net/?f=ACy%3D%28AC%29sin%2845%5C%C2%B0%29)
we have
![sin(45\°)=\frac{\sqrt{2}}{2}](https://tex.z-dn.net/?f=sin%2845%5C%C2%B0%29%3D%5Cfrac%7B%5Csqrt%7B2%7D%7D%7B2%7D)
![AC=\frac{7\sqrt{2}}{2}\ units](https://tex.z-dn.net/?f=AC%3D%5Cfrac%7B7%5Csqrt%7B2%7D%7D%7B2%7D%5C%20units)
substitute
![ACy=(\frac{7\sqrt{2}}{2})\frac{\sqrt{2}}{2}=3.5\ units](https://tex.z-dn.net/?f=ACy%3D%28%5Cfrac%7B7%5Csqrt%7B2%7D%7D%7B2%7D%29%5Cfrac%7B%5Csqrt%7B2%7D%7D%7B2%7D%3D3.5%5C%20units)
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
<em><u>Find the second possible coordinate of C</u></em>
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)
![ACx=(AC)cos(45\°)](https://tex.z-dn.net/?f=ACx%3D%28AC%29cos%2845%5C%C2%B0%29)
we have
![cos(45\°)=\frac{\sqrt{2}}{2}](https://tex.z-dn.net/?f=cos%2845%5C%C2%B0%29%3D%5Cfrac%7B%5Csqrt%7B2%7D%7D%7B2%7D)
![AC=\frac{7\sqrt{2}}{2}\ units](https://tex.z-dn.net/?f=AC%3D%5Cfrac%7B7%5Csqrt%7B2%7D%7D%7B2%7D%5C%20units)
substitute
![ACx=(\frac{7\sqrt{2}}{2})\frac{\sqrt{2}}{2}=3.5\ units](https://tex.z-dn.net/?f=ACx%3D%28%5Cfrac%7B7%5Csqrt%7B2%7D%7D%7B2%7D%29%5Cfrac%7B%5Csqrt%7B2%7D%7D%7B2%7D%3D3.5%5C%20units)
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)
![ACy=(AC)sin(45\°)](https://tex.z-dn.net/?f=ACy%3D%28AC%29sin%2845%5C%C2%B0%29)
we have
![sin(45\°)=\frac{\sqrt{2}}{2}](https://tex.z-dn.net/?f=sin%2845%5C%C2%B0%29%3D%5Cfrac%7B%5Csqrt%7B2%7D%7D%7B2%7D)
![AC=\frac{7\sqrt{2}}{2}\ units](https://tex.z-dn.net/?f=AC%3D%5Cfrac%7B7%5Csqrt%7B2%7D%7D%7B2%7D%5C%20units)
substitute
![ACy=(\frac{7\sqrt{2}}{2})\frac{\sqrt{2}}{2}=3.5\ units](https://tex.z-dn.net/?f=ACy%3D%28%5Cfrac%7B7%5Csqrt%7B2%7D%7D%7B2%7D%29%5Cfrac%7B%5Csqrt%7B2%7D%7D%7B2%7D%3D3.5%5C%20units)
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