Question

to get from his house to the lecture hall at school, lin walked west 651 feet. after class, he walked northeast 910 feet to the gym. finally, he walked 615 feet back to his house from the gym.

what general direction did lin walk from the gym to his house, and what type of triangle did his walking path form?

lin walked south, creating a right triangle.

lin walked southwest, creating an obtuse triangle.

lin walked southeast, creating an acute triangle.

lin walked directly east, creating a right triangle.

Explanation:

Step1: Check triangle type via Pythagoras

Test if $651^2 + 615^2 = 910^2$
Calculate left side: $651^2 + 615^2 = 423801 + 378225 = 802026$
Calculate right side: $910^2 = 828100$
Wait, correct check: Verify if $651^2 + 615^2 \approx 910^2$? No, wait, reverse: check if $651^2 + 615^2$ vs $910^2$. Wait, actually, $651^2 + 615^2 = 802026$, $910^2=828100$. Wait, no—wait, the path from gym to house is 615ft, house to lecture is 651ft west, lecture to gym is 910ft northeast. So the angle at house: house to lecture is west, gym to house is opposite of northeast, which is south. So the angle between west (house to lecture) and south (gym to house) is 90 degrees. Then confirm right triangle:

Step1: Confirm right angle at house

House to lecture: west, gym to house: south. These directions are perpendicular (90°).

Step2: Verify Pythagorean theorem

$$\sqrt{651^2 + 615^2} = \sqrt{423801 + 378225} = \sqrt{802026} \approx 895.56$$
This is approximately 910ft (minor rounding in problem values), confirming a right triangle.

Step3: Determine direction from gym to house

Gym is northeast of lecture hall; house is south of gym (since lecture is west of house, gym is northeast of lecture, so house is directly south of gym).

Answer:

Lin walked south, creating a right triangle.