Question

Triangle LMN
is inscribed inside a circle with diameter LM
and the inscribed angle at N
. The length of LM
is 22 cm and the length of MN
is 7 cm. Using what you know about inscribed angles, find the approximate angle measure of L
.(1 point)
Responses

38°
38°

90°
90°

71°
71°

19°
19°
Remaining Attempts : 3
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Answer 1

In a triangle inscribed in a circle where one side is the diameter, the angle opposite that side (in this case, angle N) is a right angle (90°). This is known as the Inscribed Angle Theorem.

In triangle LMN, LM is the diameter of the circle, and therefore, angle N is 90°.

To find angle L, we can use the fact that the sum of the angles in a triangle is always 180°:

\[
\text{Angle L} + \text{Angle M} + \text{Angle N} = 180°
\]

We know that angle N is 90°, so we can substitute that into the equation:

\[
\text{Angle L} + \text{Angle M} + 90° = 180°
\]

This simplifies to:

\[
\text{Angle L} + \text{Angle M} = 90°
\]

Now, the angles L and M will be complimentary. To find out the measure of angle L, we can use the Law of Cosines or the sine rule, but without calculating specifically for angle M, we can look towards the sine of the opposite side for a quick evaluation.

The triangle's side lengths are as follows:
- LM (diameter) = 22 cm
- MN = 7 cm

We can find angle L using the sine rule or trigonometric ratios. However, the key point with respect to circumference and radius leads to a simplification. Given that MN is relatively small compared to LM, angle L can be inferred.

Using a right triangle relation:

\[
\sin(\text{Angle L}) = \frac{MN}{LM} = \frac{7}{22}
\]

To calculate angle L:

\[
\text{Angle L} = \sin^{-1}\left(\frac{7}{22}\right) \approx 19.2°
\]

Thus, the closest approximate angle measure of L is

**19°**.