On a remote plateau, Emek and Tara discovered three survey markers at A(1, 2), B(7, 10), and C(13, 2) on their coordinate map. Each unit represented 1 kilometer. Before placing a temporary base, they studied the triangular region formed by the markers. Emek calculated that AB was 10 kilometers because the horizontal and vertical changes were 6 and 8. Tara found that BC was also 10 kilometers, while AC measured 12 kilometers along a horizontal line. The equal distances showed that the markers formed an isosceles triangle. They wanted the base on the 12-kilometer side because that edge followed a level ridge and provided access toward both A and C. Instead of choosing one end, Tara found the midpoint of AC at (7, 2). This placed the base 6 kilometers from both A and C. Emek also noticed that the midpoint had the same x-coordinate as B, making the layout easy to navigate. They marked (7, 2) as the base and prepared for the next stage of exploration.

Challenges

Start
Challenge 1 of 5

Challenges

Start
Challenge 1 of 4
Hint
Recall what property of side lengths defines an isosceles triangle.
A student claims that triangle ABC is isosceles because points A and C have the same y-coordinate. Which statement best evaluates this claim?
The claim is correct because any two points with the same y-coordinate form an isosceles triangle with a third point.The claim is incomplete because the triangle is isosceles specifically because AB and BC are both 10 kilometers long.The claim is correct because AC is horizontal and 12 kilometers long.The claim is incorrect because an isosceles triangle must have three equal sides.
Correct. The equal lengths AB = 10 and BC = 10 establish that the triangle is isosceles.
Not quite. Focus on the side-length condition that defines an isosceles triangle.
Next
🎉 Congrats! You got this question!
Hint
The two points have the same x-coordinate, so compare their y-coordinates.
The base is at the midpoint (7, 2) of AC, and marker B is at (7, 10). How many kilometers is the straight-line distance from the base to marker B?
Correct. The base and marker B are 8 kilometers apart vertically.
Check the difference between the y-coordinates of (7, 2) and (7, 10).
Next
🎉 Congrats! You got this question!
Hint
Average the coordinates of A and C, then compare the resulting x-coordinate with B's x-coordinate.
Emek says the base at (7, 2) is not only the midpoint of AC but also lies directly below B at (7, 10). Which calculation best verifies both parts of his statement?
The midpoint of AC is ((1 + 13) / 2, (2 + 2) / 2) = (7, 2), and this point has the same x-coordinate as B.The midpoint of AC is ((13 - 1) / 2, (2 + 2) / 2) = (6, 2), and 6 is close to B's x-coordinate.The midpoint of AC is ((1 + 13) / 2, (2 + 10) / 2) = (7, 6), and this point has the same x-coordinate as B.The midpoint of AC is (1 + 13, 2 + 2) = (14, 4), and B lies halfway between these coordinates.
Correct. The midpoint is (7, 2), and sharing x = 7 with B means the two points are vertically aligned.
Not quite. First use the midpoint formula correctly, then compare the midpoint's x-coordinate with B's.
Next
🎉🎉🎉🎉🎉🎉🎉🎉
🎉 Congrats! You finished this challenge set!
Gizli Platonun Haritasını Çıkarmak
TR
ARC_G09_GM_02_STORY_03
Gizli Platonun Haritasını Çıkarmak
Mapping the Hidden Plateau
EN
ARC_G09_GM_02_STORY_03
Mapping the Hidden Plateau