https://r2.erweima.ai/mathbot/image/17261543099322zsv69e1.jpg

Let's solve this step by step based on the image you provided. The diagram shows a circle with center \( K \), diameter \( \overline{LN} \), and angle \( \angle LKM = 85^\circ \). Here's how to fill in the blanks:

### (a) Identify an **inscribed angle**:
- An inscribed angle is one whose vertex lies on the circumference of the circle, and its sides are chords of the circle.
- In the diagram, \( \angle LPM \) is an inscribed angle because \( P \) is on the circumference, and \( \overline{LP} \) and \( \overline{PM} \) are chords.

So, the inscribed angle is **\( \angle LPM \)**.

### (b) Identify a **semicircle**:
- A semicircle is formed by a diameter of the circle. The diameter in this case is \( \overline{LN} \), as the problem specifies that \( \overline{LN} \) is the diameter.
  
Thus, the semicircle is **\( \overline{LN} \)**.

### (c) Identify a **major arc**:
- The major arc is the arc that covers more than half the circle. 
- In this case, the major arc would be the longer arc connecting points \( L \) and \( N \) that does not pass through \( M \).

Hence, the major arc is **\( \overset{\frown}{LPN} \)**.

### (d) Find the measure of **\( m \angle LNM \)**:
- Since \( \overline{LN} \) is the diameter, \( \angle LNM \) is an inscribed angle subtended by the diameter. Therefore, by the inscribed angle theorem, \( \angle LNM = 90^\circ \) because an inscribed angle subtended by a diameter is always a right angle.

Thus, \( m \angle LNM = 90^\circ \).

### (e) Find the measure of **\( m \angle LMN \)**:
- To find \( m \angle LMN \), we use the fact that the sum of the interior angles of a triangle is \( 180^\circ \).
- In triangle \( LKM \), we already know two angles: \( \angle LKM = 85^\circ \) and \( \angle LNM = 90^\circ \). So, the remaining angle is:
  \[
  m \angle LMN = 180^\circ - 85^\circ - 90^\circ = 5^\circ
  \]

Therefore, \( m \angle LMN = 5^\circ \).

### Summary of answers:
- (a) \( \angle LPM \)
- (b) \( \overline{LN} \)
- (c) \( \overset{\frown}{LPN} \)
- (d) \( m \angle LNM = 90^\circ \)
- (e) \( m \angle LMN = 5^\circ \)

Would you like more detailed explanations or further assistance with any part of the problem?

Here are 5 related questions to consider:
1. What is the relationship between a central angle and an inscribed angle subtended by the same arc?
2. How do you calculate the length of an arc in a circle?
3. What is the significance of the angle subtended by a diameter in any circle?
4. Can an inscribed angle ever be greater than \( 90^\circ \)?
5. How do you determine if a chord is a diameter based on the angles in a circle?

**Tip:** Remember that angles subtended by the same arc are equal, a useful property for many circle theorems.

En el siguiente círculo, K es el centro, LN es un diámetro y m∠LKM = 85°. Usar esta información para llenar los espacios vacíos.

Solve a geometry problem involving a circle with center K, diameter LN, and an angle LKM = 85°. This solution covers key concepts such as inscribed angles, semicircles, and major arcs. Suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation