Final answer:
The triangle is a right triangle due to the Pythagorean theorem. The perpendicular height of interest for a 13 cm base is found using the area of a right triangle and results in a length of 6/13 cm. Consequently, the value of m÷10 approximates nearest to 0, corresponding to Option A: 6.
Step-by-step explanation:
The student's question asks about finding the length of the perpendicular from the opposite vertex to the side of length 13 cm in a triangle with sides 5 cm, 12 cm, and 13 cm. Since the sides of the triangle satisfy the Pythagorean theorem (5² + 12² = 13²), we can conclude that the triangle is a right triangle, with the 5 cm and 12 cm sides being the perpendicular sides and the 13 cm side being the hypotenuse.
To find the height (m) of the triangle, we can use the formula for the area of a right triangle which is (1/2) × base × height. Knowing that the base is 12 cm and using the fact that the other side of 5 cm also serves as the height for that particular base, we find the area to be (1/2) × 12 cm × 5 cm = 30 cm². The height corresponding to the base of 13 cm, which is the length m, will give us the same area. So, (1/2) × 13 cm × m = 30 cm², solving for m will give m = (30 cm² × 2)/13 cm = 60/13 cm.
To find the value of m÷10, that is m divided by 10, we calculate (60/13) cm ÷ 10 = 6/13 cm. This approximates to the nearest whole number, which is 0 since fractions of centimeters are not considerations in the options provided. Hence, the closest answer choice is Option A: 6.