Math Problem Statement

The image shows a geometric problem involving a semicircle and triangles. The questions are: 1) What type of triangle is △OAB? 2) What is the measure of the central angle labeled ⑦?

Solution

The image shows a geometric problem involving a semicircle and triangles. Let's analyze the content and answer the questions.

Problem Analysis:

  1. 問1: What type of triangle is △OAB?
  2. 問2: What is the measure of the central angle labeled ⑦?

Solution:

  1. 問1 (Type of △OAB):

    • The figure shows a semicircle with diameter OBOB. Triangle OABOAB is inscribed in the semicircle with OBOB as its hypotenuse.
    • By the Inscribed Angle Theorem, any triangle inscribed in a semicircle with the diameter as one side is a right triangle.
    • Therefore, △OAB is a right triangle with a right angle at point AA.
  2. 問2 (Measure of the central angle labeled ⑦):

    • Central angle ⑦ is subtended by arc ABAB of the semicircle. In a semicircle, the central angle that subtends the entire diameter is always 180180^\circ. However, since this question specifically refers to angle ⑦, which likely subtends only a segment of the semicircle:
    • The measure of angle ⑦ depends on the specific configuration shown. Without precise measurements, it's challenging to provide an exact degree, but the information suggests it is proportional to the arc's length that angle ⑦ subtends.

Would you like more details on the central angle calculation, or do you have any other questions?

Related Questions:

  1. How do you determine the type of a triangle inscribed in a semicircle?
  2. What is the Inscribed Angle Theorem, and how does it apply to triangles in a semicircle?
  3. How do you calculate central angles in circles when given arcs?
  4. What properties do right triangles have when inscribed in semicircles?
  5. Can the measure of angle ⑦ be determined exactly without additional data?

Tip: Always remember that a triangle with one side as the diameter of a semicircle is a right triangle, a key property of semicircle inscribed figures.

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Math Problem Analysis

Mathematical Concepts

Geometry
Triangles
Semicircles
Angles

Formulas

Inscribed Angle Theorem
Central Angle Formula

Theorems

Inscribed Angle Theorem
Right Triangle Property in a Semicircle

Suitable Grade Level

Grades 8-10