### Video Transcript

Consider the orthonormal coordinate system π; πΊ, πΉ. Find the coordinates of point π΄, find the coordinates of point πΊ, and find the coordinates of point π».

Letβs begin by recalling the definition of an orthonormal coordinate system. This is a coordinate plane where the π₯- and π¦-axes are perpendicular and the unit lengths are equal in the π₯- and π¦-directions. The first letter in the coordinate system π; πΊ, πΉ represents the origin. And this means that π has coordinates zero, zero. The second letter πΊ means that line ππΊ is the π₯-axis, with ππΊ as its unit length. This means that πΊ has coordinates one, zero. In the same way, line ππΉ is the π¦-axis, with ππΉ as its unit length. And as such, the coordinates of point πΉ are zero, one.

In the three parts to this question, we are asked to find the coordinates of points π΄, πΊ, and π». As already mentioned, point πΊ has coordinates one, zero. Since π΄π΅πΆπ· is a square, the length of ππΉ is equal to the length of ππ». This means that point π» is the same distance below the π₯-axis as point πΉ is above the π₯-axis. And since πΉ has coordinates zero, one, π» has coordinates zero, negative one.

Using the symmetry of our diagram once again, we see that point πΈ has coordinates negative one, zero. This is because it is the same distance in the negative π₯-direction from the origin than point πΊ is in the positive π₯-direction. Drawing lines perpendicular to the π₯- and π¦-axes, we see that point π΄ has an π₯-coordinate equal to negative one and a π¦-coordinate equal to one. And we can therefore conclude that the coordinates of point π΄ are negative one, one.

Whilst it is not required in this question, we can also establish that point π΅ has coordinates one, one; point πΆ has coordinates one, negative one; and point π· has coordinates negative one, negative one. The three answers to this question, however, are point π΄ is equal to negative one, one; point πΊ is equal to one, zero; and point π» is equal to zero, negative one.