Assume triangle JKL is in the first quadrant, with the measure of angle K = 90°. Suppose triangle JKL is a 30°-60°-90° triangle and segment JK is the side opposite the 60° angle. What are the approximate coordinates of point L?
j(2,7)
k(2,2)
a(4.9,2)
b(4.5,2)
c(7.1,2)
d(8.7,2)
4.9,2
To find the approximate coordinates of point L, we need to use the information given in the problem.
Since triangle JKL is a 30°-60°-90° triangle and angle K = 90°, we can determine the lengths of the sides.
The side opposite the 60° angle, which is segment JK, has a length of 5 units.
Since point J has coordinates (2,7) and segment JK lies on the x-axis, we can determine the x-coordinate of point L.
To find the x-coordinate of point L, we can add the length of segment JK (5 units) to the x-coordinate of J.
x-coordinate of L = x-coordinate of J + length of JK = 2 + 5 = 7.
Now, we know that the x-coordinate of point L is 7.
Since triangle JKL is in the first quadrant and angle K = 90°, the y-coordinate of point L can be determined as the y-coordinate of K.
y-coordinate of L = y-coordinate of K = 2.
Therefore, the approximate coordinates of point L are (7,2). Hence, the correct answer is option b) (4.5,2).
To find the approximate coordinates of point L, we can use the given coordinates of points J and K, along with the properties of a 30°-60°-90° triangle.
In a 30°-60°-90° triangle, the side opposite the 30° angle is half the length of the hypotenuse, and the side opposite the 60° angle is √3 times the length of the side opposite the 30° angle.
In this case, segment JK is the side opposite the 60° angle.
Given:
J(2,7)
K(2,2)
Since segment JK is the side opposite the 60° angle, the length of JK is equal to √3 times the length of the side opposite the 30° angle.
To find the length of JK, we can calculate the vertical component (difference in y-coordinates) of points J and K.
JK = K y-coordinate - J y-coordinate
= 2 - 7
= -5
To find the length of KL, we can multiply JK by √3.
KL = JK * √3
= -5 * √3
Now, to find the coordinates of point L, we add the horizontal component (same x-coordinate as K) and the vertical component (calculated above) to the coordinates of point K.
L x-coordinate = K x-coordinate = 2
L y-coordinate = K y-coordinate + KL = 2 - 5 * √3
The approximate coordinates of point L are (2, 2 - 5 * √3).
So, the answer is:
b(4.5, 2)