Final answer:
To graph a right triangle and find the distance between two points (7,1) and (-2,6) in simplest radical form, we use the Pythagorean theorem. The horizontal and vertical distances between the points are 9 units and 5 units respectively. The length of the hypotenuse is √106 units.
Step-by-step explanation:
To graph a right triangle with the hypotenuse connecting the points (7,1) and (-2,6), we first plot these points on a coordinate plane. Then, we find the distances between these points along the x-axis and y-axis, which will be the lengths of the two legs of the triangle.
The horizontal distance (x-axis) is the absolute difference between the x-coordinates of the two points: |7 - (-2)| = 9 units. The vertical distance (y-axis) is the absolute difference between the y-coordinates of the two points: |1 - 6| = 5 units. These distances form the legs of the right triangle.
To find the distance between the two points or the length of the hypotenuse, we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). This is given by the formula c = √(a² + b²).
Thus, the length of the hypotenuse is: √(9 units)² + (5 units)² = √81 + 25 = √106 which cannot be simplified further. So the distance between the two points in simplest radical form is √106 units.