A ball game between Dylan and Mya is about to begin. Using sidewalk chalk, they draw their “arena” on the ground. The arena can be modeled by the equation x/2 + y/2 = 81 . Dylan has taken his position at (-5, -2) and Mya has taken her position at (6, 4). Each graph unit represents 1 meter.
what are the points
How far apart are Dylan and Mya (to two decimal places)?
To find the distance between two points, we can use the distance formula.
The distance formula is given by:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Given the points (x1, y1) = (-5, -2) and (x2, y2) = (6, 4), we can substitute these values into the distance formula:
d = sqrt((6 - (-5))^2 + (4 - (-2))^2)
Simplifying further:
d = sqrt(11^2 + 6^2)
d = sqrt(121 + 36)
d = sqrt(157)
d ≈ 12.53 meters
Therefore, Dylan and Mya are approximately 12.53 meters apart.
Jordyn joins the game so that she forms a triangle with Dylan and Mya. Her exact coordinates are NOT important, but we do know that from Jordyn’s position, the angle formed to Dylan and Mya is 76°. From Dylan’s position, the angle formed to Mya and Jordyn is 50°.
Draw a diagram below with this information and the information from Tasks 1 and 2.
Use this diagram to find the distances from Jordyn to Dylan and Jordyn to Mya (to two decimal places).
Apologies, but I'm unable to draw a diagram for you. However, I can provide you with the steps to determine the distances from Jordyn to Dylan and from Jordyn to Mya.
Let's assume that the position of Jordyn is denoted by the point (x3, y3).
To find the distance from Jordyn to Dylan, we can use the distance formula:
d(Dylan, Jordyn) = sqrt((x3 - (-5))^2 + (y3 - (-2))^2)
To find the distance from Jordyn to Mya, we can also use the distance formula:
d(Mya, Jordyn) = sqrt((x3 - 6)^2 + (y3 - 4)^2)
Now, we need to use the information regarding angles to find the relationship between the distances. Given that the angle formed at Dylan's position is 50° and the angle formed at Jordyn's position is 76°, we can use the Law of Sines to set up a ratio:
sin(50°) / d(Dylan, Mya) = sin(76°) / d(Dylan, Jordyn)
Therefore, we can rearrange the equation to solve for d(Dylan, Jordyn):
d(Dylan, Jordyn) = (sin(50°) * d(Mya, Jordyn)) / sin(76°)
Using the above equation, we can find the distance from Jordyn to Dylan.
Similarly, we can solve for d(Mya, Jordyn) using the Law of Sines:
sin(76°) / d(Mya, Jordyn) = sin(50°) / d(Dylan, Jordyn)
Rearranging the equation above, we can solve for d(Mya, Jordyn).
Please note that to calculate the exact distances, we would need the coordinates of Jordyn's position.