### Video Transcript

Consider points π΄ one, one; π΅ negative one, one; and πΆ negative one, three in the orthonormal coordinate plane π; πΌ, π½. What type of coordinate plane is π; π΄, π΅? What are the coordinates of point πΆ in the coordinate plane π; π΄, π΅.

The first part of this question asked us to determine what type of coordinate plane π; π΄, π΅ is. The first letter, in this case π, corresponds to the origin. So π has coordinates zero, zero. Line ππ΄ is the π₯-axis with ππ΄ as its unit length. And line ππ΅ is the π¦-axis with ππ΅ as its unit length. To identify the type of coordinate plane, we need to determine two things: firstly, where the lines ππ΄ and ππ΅ are perpendicular and, secondly, whether the length of ππ΄ is equal to the length of ππ΅.

We notice from the figure that line segments ππ΄ and ππ΅ are both diagonals of a square on the grid. We know that the diagonal of a square is an axis of symmetry of the square. This means that the measure of angle π΄ππΌ is equal to the measure of angle π΄ππ½. And both of these are equal to 45 degrees. In the same way, the measure of angle π΅ππ½ is 45 degrees. And since 45 plus 45 is equal to 90, the measure of angle π΄ππ΅ is 90 degrees. And we can conclude that lines ππ΄ and ππ΅ are perpendicular.

We also note from the figure that as the diagonals of a square are of equal lengths and line segments ππ΄ and ππ΅ are diagonals of two congruent squares, we have ππ΄ is equal to ππ΅. Since the answer to both of our questions is yes, we can conclude that the plane π; π΄, π΅ is an orthonormal coordinate plane since its axes are perpendicular and they have the same unit length.

The second part of this question asks us to find the coordinates of point πΆ in this coordinate plane. Note that this is not the same as the coordinate of negative one, three as given in the question since these are the coordinates of point πΆ in the plane π; πΌ, π½. We can do this by firstly drawing two lines that are parallel to the π₯- and π¦-axes passing through point πΆ. By the definition of the coordinate plane π΄; π, π΅, the length of line segment ππ΄ is one unit in the π₯-direction. And hence, the coordinates of point π΄ are one, zero. In the same way, the line segment ππ΅ is one unit in the π¦-direction. And π΅ has coordinates zero, one.

Since the line parallel to the π¦-axis passing through πΆ intersects the π₯-axis at π΄, we have an π₯-coordinate of one. The line parallel to the π₯-axis passing through πΆ intersects the π¦-axis at a point that is a distance from the origin twice the length of ππ΅. Since this is on the positive side of the π¦-axis, i.e, the same side as π΅, this corresponds to a π¦-coordinate of two. The coordinates of point πΆ in the coordinate plane π; π΄, π΅ are one, two.