In rectangle JNML, where JN = 4x + 8, LN = 5x + 1, and ∠JNM = 124°, we found that:
KM = √(-16x^2 + 70x + 63)
∠NKL = arcsin(√(-16x^2 + 70x + 63) / (5x + 1))
These expressions depend on the variable x, so the lengths and measures will vary depending on the specific value of x.
We are given a rectangle JNML, where:
JN = 4x + 8
LN = 5x + 1
∠JNM = 124°
We are asked to find:
KM
∠NKL
Step 1: Find JM
Since JNML is a rectangle, we know that JM = LN. Substituting the given expression for LN, we get:
JM = 5x + 1
Step 2: Find KM using the Pythagorean Theorem
Triangle JKM is a right triangle, where JM is the hypotenuse and KM and JN are the legs. We can use the Pythagorean theorem to solve for KM:
KM^2 = JN^2 - JM^2
Substituting the given expressions for JN and JM, we get:
KM^2 = (4x + 8)^2 - (5x + 1)^2
Expanding the squares and combining like terms, we get:
KM^2 = 9x^2 + 80x + 64 - 25x^2 - 10x - 1
KM^2 = -16x^2 + 70x + 63
Taking the square root of both sides, we get:
KM = ±√(-16x^2 + 70x + 63)
Since KM must be a positive length, we take the positive square root:
KM = √(-16x^2 + 70x + 63)
Step 3: Find ∠NKL
Triangle NKL is also a right triangle, where ∠NKL is the opposite angle to KM. We can use the sine function to solve for ∠NKL:
sin(∠NKL) = KM / JM
Substituting the expressions for KM and JM, we get:
sin(∠NKL) = √(-16x^2 + 70x + 63) / (5x + 1)
∠NKL = arcsin(√(-16x^2 + 70x + 63) / (5x + 1))