A coordinate proof is a type of geometric proof in which we visualize the given information in a coordinate plane and then coordinates, formulas, and rules to prove a statement. In this case we are told that we are camping with our cousin. For simplicity, let's place the campsite at the origin O of the coordinate plane.
We know that we hike to a point that is 500 meters east and 1200 meters north of a campsite. Let's add our position to the coordinate plane. We will name this point A.
Next, we are told that our cousin hikes to a point that is 1000 meters east of the campsite. Let's add this information to our diagram and label the point that represents our cousin's position B.
Let's connect the drawn points to form a triangle.
To prove that the triangle is isosceles, we need to calculate the lengths of its three sides and see whether at least two of them are congruent. To do this we will use the Distance Formula.
d = sqrt((x_2-x_1)^2 + (y_2-y_1)^2)
Side
Points
Substitute
Simplify
AO
A( 500,1200) and O( 0,0)
AO = sqrt(( 0- 500)^2+( 0- 1200)^2)
AO=1300
BA
B( 1000,0) and A( 500,1200)
BA = sqrt(( 500- 1000)^2+( 1200- 0)^2)
BA=1300
BO
B( 1000,0) and O( 0,0)
BO = sqrt(( 0- 1000)^2+( 0- 0)^2)
BO=1900
Since the length of AO is equal to the length of BA, the triangle formed by our position, our cousin's position, and the campsite is isosceles. Notice that the position of the triangle in the coordinate plane does not affect the final result.
