Application of Distance, Midpoint, and Slope — Quiz
Question 10 • Geometry with Statistics
A graph shows the first triangle in a wallpaper pattern. The coordinates are A (−4,−8) , B (−7,4) , and C (6,5) How can the distance formula be used to find the side lengths and classify the triangle?
Answer
A
The side lengths are square root of 137, square root of 82, and square root of 13, so this is an isosceles triangle.
B
The side lengths are 317 , square root of 170, and square root of 269, so this is an isosceles triangle.
C
The side lengths are 3 square root of 17, square root of 170, and square root of 269, so this is a scalene triangle.
D
The side lengths are square root of 137, square root of 82, and square root of 13, so this is a scalene triangle.
Explanation
Use the distance formula for each pair of vertices. For AB, the distance is √[(-7 − (−4))² + (4 − (−8))²] = √153 = 3√17; for BC, it is √[(6 − (−7))² + (5 − 4)²] = √170. For CA, it is √[(-4 − 6)² + (−8 − 5)²] = √269. Since all three side lengths are different, the triangle is scalene, matching choice C.
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