Question

Use the image to answer the question.

A circle has center point upper F. The circumference point upper L is at the left of the circle and the circumference point upper M is at the right of the circle. Point upper K is above the circle in line with upper F.

If KL=3x+2
and KM=14 cm
, then what is the value of x?

(1 point)

Answer 1

Given the information from the problem, we can establish that points K, L, and M form a vertical line where K is above the center F of the circle, and L and M are on the circumference of the circle.

Since KL and KM represent distances from point K to points L and M respectively, we can express the distances as follows:

- KL = distance from K to L = \( 3x + 2 \)
- KM = distance from K to M = \( 14 \, \text{cm} \)

From the arrangement of points, we know that:

\[
KL + LM = KM
\]

Since L and M are on opposite sides of point K, we can deduce that:

\[
LM = KL + KM
\]

We can set up the equation based on the distances mentioned. Because LM is the distance between points L and M, we have:

\[
KL + LM = KM
\]

Rearranging the equation to find the distances:

\[
LM = KM - KL
\]

Substituting in the known values gives us:

\[
LM = 14 - (3x + 2)
\]

\[
LM = 14 - 3x - 2
\]

\[
LM = 12 - 3x
\]

Now looking at the entire distance LM and the fact it is equal to the difference of distances:

Since KL + LM = KM, we set those equal:

\[
(3x + 2) + (12 - 3x) = 14
\]

This simplifies and allows us to solve for x:

\[
3x + 2 + 12 - 3x = 14
\]

The \(3x\) cancels out:

\[
14 = 14
\]

This true statement does not help us find x directly.

Therefore, we solve for x by using the shorter method as KL and KM are both heights.

Thus, let’s solve for x using only KM = KL + LM where LM = 12 - 3x:

We know the circle's symmetry results in equal halves so:

Since total distance would calculate KL + LM as equal to the other side, it shows symmetry implies that:

So we need to find the discrepancy.

We adjust and state:

Since both total was equal to KL + 3x part equaling altogether:

We would say since 3x + 2 = 14 directly leads to:

\[
3x + 2 = 14
\]

Moving 2 across gives:

\[
3x = 12
\]

\[
x = 4
\]

Thus, the value of x is:

\[
\boxed{4}
\]