Segment AB is the hypotenuse of the right isosceles ΔABC with A(–6, –2) and B(–6, 5). Find all possible coordinates of C.

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 thatTriangle ABC is a right isosceles trianglesoIs a 45°-90°-45° triangleAC=BCwe haveA(-6,-2), B(-6,5)step 1Find the length side of the hypotenuse AB[tex]AB=5-(-2)=7\ units[/tex]step 2Applying the Pythagoras TheoremFind the length side of leg AC[tex]AB^{2}=AC^{2}+BC^{2}[/tex]Remember thatAC=BCsubstitute 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 3Find the first possible coordinates of CThe point C is located at right of point ADetermine the x-coordinate of point CThe x-coordinate of point C must be equal to the x-coordinate of point A plus the horizontal distance between point A and point CLetACx ------> the horizontal distance between point A and point CThe 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 isCx=-6+3.5=-2.5 Determine the y-coordinate of point CThe y-coordinate of point C must be equal to the y-coordinate of point A plus the vertical distance between point A and point CLetACy ------> the vertical distance between point A and point CThe 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 isCy=-2+3.5=1.5 thereforeThe possible first coordinates of point C are (-2.5,1.5)step 4Find the second possible coordinate of CThe point C is located at left of point ADetermine the x-coordinate of point CThe x-coordinate of point C must be equal to the x-coordinate of point A minus the horizontal distance between point A and point CLetACx ------> the horizontal distance between point A and point CThe 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 isCx=-6-3.5=-9.5 Determine the y-coordinate of point CThe y-coordinate of point C must be equal to the y-coordinate of point A plus the vertical distance between point A and point CLetACy ------> the vertical distance between point A and point CThe 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 isCy=-2+3.5=1.5 thereforeThe possible second coordinates of point C are (-9.5,1.5)see the attached figure to better understand the problem