### Video Transcript

π΄, π΅, and πΆ are three points plotted on a grid. There are three parts to this question. A) Plot the point π· on the grid such that π΄π΅πΆπ· is a square. b) The point πΈ is the midpoint of π΄π΅. Circle the two names for triangle π΅πΆπΈ: right-angled, scalene, isosceles, equilateral. Part c) Circle the ratio which is equivalent to the area of triangle π΅πΆπΈ to the area of square π΄π΅πΆπ·. Is it one to two, one to three, one to four, or one to eight?

The coordinates of the three points already on the grid are as follows. π΄ has coordinates seven, eight. π΅ is nine, three. And πΆ is four, one. Weβre told that plotting point π· on the grid creates a square, π΄π΅πΆπ·. As all four sides of a square are equal in length, the length of π΅π΄ and π΅πΆ will be equal to πΆπ· and π·π΄. Line π΄π΅ will also be parallel to line πΆπ·, and line π΅πΆ will be parallel to line π΄π·.

Letβs first consider how we get from point π΅ to point π΄. To get from point π΅ to point π΄, we move five squares up and two squares to the left. This is because the difference in the π¦-coordinates is five, and the difference between the π₯-coordinates is two. As line π΅π΄ is parallel to line πΆπ·, we must also move five squares up and two squares to the left to get from πΆ to π·. This means that point π· will have coordinates two, six.

Completing the square, we can now see that π΅π΄ is parallel to πΆπ· and π΄π· is parallel to π΅πΆ. All four sides of the shape π΄π΅πΆπ· are equal. Therefore, it is a square. The point π· on the grid such that π΄π΅πΆπ· is a square has coordinates two, six.

The second part of our question tells us that point πΈ is the midpoint of π΄π΅. To work out where the midpoint of any two coordinates is, we add the two coordinates and divide by two. The π₯-coordinates of π΄ and π΅ are seven and nine. Seven plus nine is equal to 16. Dividing this by two gives us eight. This means that the π₯-coordinate of point πΈ is eight. The π¦-coordinates of π΄ and π΅ are eight and three. We add these and divide by two. Eight plus three is equal to 11. Dividing this by two gives us 5.5. This means that the π¦-coordinate of πΈ is 5.5. The point πΈ, which is the midpoint of π΄π΅, has coordinates eight, 5.5.

We were asked to consider the two names for triangle π΅πΆπΈ as shown in pink. As the angle at π΅ is the angle within the square, we know it is right-angled. Therefore, one name for triangle π΅πΆπΈ is right-angled. The other three types of triangles have the following properties. An equilateral triangle has three equal sides and three equal angles. An isosceles triangle has two equal sides and two equal angles. A scalene triangle has no equal sides and no equal angles.

As the initial shape π΄π΅πΆπ· was a square, we know that the length π΅πΆ is equal to the length π΅π΄. This means that the length π΅πΈ is half of the length π΅πΆ. This means we can rule out equilateral triangle. The longest side of a right-angled triangle is known as the hypotenuse. This is longer than both of the other sides. Therefore, we can rule out isosceles. All three sides of the triangle π΅πΆπΈ are different lengths. Therefore, the triangle is scalene. The two names for triangle π΅πΆπΈ are right-angled and scalene.

The third part of the question asked us to consider the ratio of the area of the triangle π΅πΆπΈ to the area of the square π΄π΅πΆπ·. Letβs first let the length of each side of the square be letter π₯. This means that the length π΅πΈ will be a half π₯, as πΈ was the midpoint of π΄π΅. The area of the square π΄π΅πΆπ· will therefore be equal to π₯ squared, as the area of any square is equal to one length squared.

The area of any triangle is equal to a half multiplied by the base multiplied by the height. In this case, the area of triangle π΅πΆπΈ is equal to a half multiplied by π₯ multiplied by a half π₯. One-half multiplied by one-half is one-quarter, and π₯ multiplied by π₯ is equal to π₯ squared. Therefore, the area of the triangle π΅πΆπΈ is equal to one-quarter π₯ squared. This means that the area of the triangle is a quarter of the area of the square.

We can show this on the grid by splitting the square into four identical triangles. Four triangles the same size as π΅πΆπΈ are the same area as the square π΄π΅πΆπ·. This means that the ratio of the triangle to the square is one to four. The ratio that is equivalent to area of triangle π΅πΆπΈ to area of square π΄π΅πΆπ· is one to four.