Final answer:
To find the length of one of the legs of a right triangle with vertices on a coordinate plane, the Pythagorean theorem is used. The hypotenuse is vertical with a length of 13 units, so one leg is 13 units long as well, making option C) 13 the correct answer.
Step-by-step explanation:
The student asked about the length of one leg of a right triangle on a coordinate plane, which has a hypotenuse defined by vertices at (0,2) and (0,15). To solve this problem, we can use the Pythagorean theorem which states that for a right triangle with legs of length a and b, and hypotenuse of length c, the relationship is a² + b² = c². Since the hypotenuse's vertices are only varying in the y-coordinate, it is vertical, and its length is the difference in the y-values which is |15 - 2| = 13 units.
Given that the lengths of the two legs are whole number values, and one leg will be along the x-axis and the other along the y-axis due to the nature of the coordinates given, we can surmise the length of one of the legs by testing whole number values that when squared and added together equal 13².
Therefore, the length of one of the legs can be found by trying the whole number options provided: (A) 10, (B) 12, (C) 13, or (D) 14, to see which squared value, when added to the square of the other leg, equals to 13². In this case, leg a would be along the x-axis and leg b would be 13 units along the y-axis as derived from the vertices.
C is the correct choice because 0² + 13² = 13². This means one of the legs of the triangle, perpendicular to the hypotenuse, is 13 units long.