### Video Transcript

Point negative six, seven is on the circle with center negative seven, negative one. Decide whether the point negative eight, negative nine is on, inside, or outside the circle.

We are told in the question that the point with coordinates negative six, seven lies on a circle with center negative seven, negative one as shown. And we need to decide whether the point with coordinates negative eight, negative nine lies on the circle, is inside the circle, or is outside the circle. We will begin by calculating the length of the radius of the circle. This is the distance between the points negative six, seven and negative seven, negative one. We will then work out the distance between the point negative eight, negative nine and the center of the circle.

If we let this distance equal 𝑑, then if 𝑑 is equal to the radius 𝑟, the point lies on the circle. If 𝑑 is less than 𝑟, the point lies inside the circle. And finally, if 𝑑 is greater than 𝑟, the point lies outside the circle. To calculate the length of the radius 𝑟, we will use the Pythagorean theorem. This states that in any right triangle 𝑎 squared plus 𝑏 squared is equal to 𝑐 squared, where 𝑐 is the length of the longest side or hypotenuse and 𝑎 and 𝑏 are the lengths of the other two sides.

In the right triangle drawn, the lengths of the shorter sides are one and eight units. This is because the difference between our 𝑥- and 𝑦-coordinates are one and eight units, respectively. Substituting these values into the Pythagorean theorem, we have 𝑟 squared is equal to one squared plus eight squared. One squared is equal to one and eight squared is 64. Therefore, 𝑟 squared is equal to 65. Taking the square root of both sides of this equation and since 𝑟 must be positive, the radius is equal to root 65 units.

As already mentioned, our next step is to calculate the distance between the center of the circle and the point with coordinates negative eight, negative nine. The difference between the 𝑥-coordinates here is also one and the difference between the 𝑦-coordinates is once again eight. This means that 𝑑 squared is also equal to one squared plus eight squared. The distance 𝑑 between the point negative eight, negative nine and the center of the circle is also root 65 units.

We can therefore conclude that since this is equal to the radius, the point negative eight, negative nine lies on the circle.