Question

Сформулювати і довести наслідки з теореми провписаний кут.даю 20 балов​

Answer 1

Відповідь:

Пояснення:

  Наслідок 1 .  Вписані кути, що спираються на одну й ту саму

дугу, рівні.

          Справді, за теоремою про вписаний кут градусна міра кожного з вписаних кутів дорівнює половині дуги AB.

Наслідок 2. Вписаний кут, що спирається на півколо,— прямий,

і навпаки: будь-який прямий вписаний кут спирається на півколо.

Наслідок 3 . Центром кола, описаного навколо прямокутного трикутника, є середина гіпотенузи . Медіана прямокутного

трикутника, проведена з вершини  прямого кута, дорівнює половині гіпотенузи.

Answer 2

The Inscribed Angle Theorem and its Consequences

The Inscribed Angle Theorem is a fundamental result in geometry that relates the measure of an inscribed angle in a circle to the measure of the intercepted arc. Let's formulate and prove the theorem, and then discuss its consequences.

Theorem: Inscribed Angle Theorem In a circle, an inscribed angle is equal in measure to half the measure of its intercepted arc.

Proof: Consider a circle with center O, and let A, B, and C be three points on the circle that form an inscribed angle ∠ABC. Let the intercepted arc be denoted by AC.

To prove the Inscribed Angle Theorem, we need to show that the measure of ∠ABC is equal to half the measure of arc AC.

We can start by drawing radii OA and OC. Since OA and OC are radii of the same circle, they are congruent. Therefore, we have OA = OC.

Next, draw segment OB. Since OB is a chord of the circle, it divides the circle into two arcs: arc AC and arc BC.

Now, let's consider triangle OAB. By the Triangle Sum Theorem, the sum of the angles in a triangle is 180 degrees. Therefore, we have:

∠OAB + ∠OBA + ∠BOA = 180 degrees

Since OA = OC, angles ∠OAB and ∠OBA are congruent. Let's denote their measure as x. Therefore, we have:

x + x + ∠BOA = 180 degrees 2x + ∠BOA = 180 degrees ∠BOA = 180 degrees - 2x

Now, let's consider arc AC. By the definition of an inscribed angle, the measure of ∠ABC is equal to half the measure of arc AC. Therefore, we have:

∠ABC = 0.5 * arc AC

Since arc AC is the difference between the measures of the entire circle and arc BC, we have:

arc AC = 360 degrees - arc BC

Substituting this into the equation above, we get:

∠ABC = 0.5 * (360 degrees - arc BC) ∠ABC = 180 degrees - 0.5 * arc BC

Comparing this with the equation for ∠BOA, we can see that ∠ABC and ∠BOA have the same measure:

∠ABC = ∠BOA

Therefore, we have proved that the measure of an inscribed angle is equal to half the measure of its intercepted arc.

Consequences of the Inscribed Angle Theorem: The Inscribed Angle Theorem has several important consequences in geometry. Here are a few notable ones:

1. Central Angle Theorem: A central angle in a circle is equal in measure to its intercepted arc. This is a direct consequence of the Inscribed Angle Theorem, as a central angle can be considered as an inscribed angle with an intercepted arc equal to the entire circle.

2. Angles Inscribed in the Same Arc: If two angles are inscribed in the same arc, they are congruent. This follows from the fact that both angles intercept the same arc, and the Inscribed Angle Theorem states that the measure of an inscribed angle is equal to half the measure of its intercepted arc.

3. Angles Inscribed in Congruent Arcs: If two arcs in a circle are congruent, then any angles inscribed in those arcs are congruent. This can be proven by considering the inscribed angles formed by the congruent arcs and applying the Inscribed Angle Theorem.

These are just a few consequences of the Inscribed Angle Theorem. The theorem itself and its consequences have wide-ranging applications in geometry and trigonometry.

I hope this explanation helps! Let me know if you have any further questions.