Question

JL is a common tangent to circles M and K at point J. If angle MLK measures 61ᵒ, what is the length of radius MJ? Round to the nearest hundredth. (Hint: Show that triangles LMJ and LKJ are right triangles, and then use right triangle trigonometry to solving for missing sides of the right triangles.)

Answer 1

To find the length of radius MJ, we can use right triangle trigonometry. Firstly, we can show that triangles LMJ and LKJ are right triangles. Then, we can use the given angle MLK of 61ᵒ to find the length of radius MJ, using the sine function. The equation to find MJ is MJ = rM * sin(29ᵒ).

Step-by-step explanation:

To find the length of radius MJ, we can use right triangle trigonometry. Firstly, we can show that triangles LMJ and LKJ are right triangles. Since JL is a common tangent, it is perpendicular to the radii of the circles at points J. Therefore, angle LMJ and angle LKJ are right angles. Now, we can use the given angle MLK of 61ᵒ to find the length of radius MJ.

Let's call the radius of circle M rM and the radius of circle K rK. In triangle LMJ, we have the following relationships:

1. angle LMJ = 90ᵒ (since it is a right triangle)
2. angle MLJ = angle MLK - angle JLK = 61ᵒ - 90ᵒ = -29ᵒ (since angle JLK is a right angle)
3. angle MJL = angle JML = 90ᵒ - angle MLJ = 90ᵒ - (61ᵒ - 90ᵒ) = 119ᵒ

Using the sine function, we can find the length of side MJ:

sin(angle MLJ) = length of side MJ / length of side LJ

sin(-29ᵒ) = MJ / rM

Since sin(angle MLJ) = -sin(angle MJL), we can rewrite the equation as:

sin(29ᵒ) = MJ / rM

Now, we can rearrange the equation to solve for MJ:

MJ = rM * sin(29ᵒ)

Since we are not given the values of rM or rK, we cannot find the specific value of MJ. However, we can use this equation to find the length of radius MJ if we are given the values of the radii of the circles and the given angle MLK.

Remember to round the answer to the nearest hundredth as specified in the question.

Answer 2

In the problem of insufficient data quantities. I can get a general solution.We know that tangent to a circle is perpendicular to the radius at the point of tangency. It's mean that triangles LJM and LJK are rights.

Let angle JLK like X.

So, angle JLM=61-x.

And it's mean that by using right triangle trigonometry

Radius MJ = LM*cos(61-X)