Question

A map of an obstacle course is shown in the graph. The running path for the course is shaped like a right triangle where each unit is equal to 1 meter.

graph of a right triangle with points at negative 6 comma 0 labeled Obstacle 1, negative 6 comma 8 labeled Starting Point, and 0 comma 0 labeled Obstacle 2

Part A: Find the distance in meters from the starting point to obstacle 2. Show every step of your work. (3 points)

Part B: How many meters is one full lap around the course? Show every step of your work. (1 point)

Answer 1

The distance from the starting point to obstacle 2 on the map, using the Pythagorean theorem, is calculated to be 10 meters. The total distance for one lap around the obstacle course, by adding the lengths of all sides of the triangle, is found to be 24 meters.

Step-by-step explanation:

Part A: Distance from Starting Point to Obstacle 2

To find the distance from the starting point to obstacle 2, we need to use the Pythagorean theorem since the running path forms a right triangle. The two legs of the right triangle are the horizontal distance from Obstacle 1 to Obstacle 2 and the vertical distance from the Starting Point to Obstacle 1.

Horizontal distance (base of the triangle) = distance between Obstacle 1 and Obstacle 2 = 6 meters (since it goes from -6 to 0 on the x-axis).

Vertical distance (height of the triangle) = distance from Starting Point to Obstacle 1 = 8 meters (since it goes from 0 to 8 on the y-axis).

Applying the Pythagorean theorem: a² + b² = c²

a = 6 meters (horizontal distance)
b = 8 meters (vertical distance)
c = ? (distance from Starting Point to Obstacle 2)

So we have:
6² + 8² = c²36 + 64 = c²100 = c²

By taking the square root of both sides we find that c = 10 meters.

Therefore, the distance from the starting point to obstacle 2 is 10 meters.

Part B: Total Distance for One Lap

To find the total distance for one lap around the course, we add up the lengths of all three sides of the triangle.

Distance from Starting Point to Obstacle 2 = 10 meters (from Part A),
Distance from Starting Point to Obstacle 1 = 8 meters (vertical leg),
Distance from Obstacle 1 to Obstacle 2 = 6 meters (horizontal leg).

Total distance for one lap = 10 + 8 + 6 = 24 meters.

Thus, one full lap around the course is 24 meters.

Answer 2

Let's calculate the distance from the starting point to obstacle 2.

Part A:
The distance (d) can be found using the distance formula: \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)

For obstacle 2 (0, 0) and the starting point (-6, 8):
\[d = \sqrt{(0 - (-6))^2 + (0 - 8)^2}\]

\[d = \sqrt{(6)^2 + (-8)^2}\]

\[d = \sqrt{36 + 64}\]

\[d = \sqrt{100}\]

\[d = 10\]

So, the distance from the starting point to obstacle 2 is 10 meters.

Part B:
Since the running path forms a right triangle, and we've found the length of two sides (6 and 8), you can use the Pythagorean theorem to find the hypotenuse, which is the distance for one full lap.

\[c = \sqrt{a^2 + b^2}\]

\[c = \sqrt{6^2 + 8^2}\]

\[c = \sqrt{36 + 64}\]

\[c = \sqrt{100}\]

\[c = 10\]

Therefore, one full lap around the course is 10 meters.