contestada

1.This figure shows circle O with chords AC and BD . ​ mAB=34∘ mCD=34∘ AP=6 m ​ PC=8 m What is BD ? Enter your answer in the box.
2.This figure shows circle C with diameter LM and inscribed ∠LMN . 

m∠LMN=32°

 What is the measure of  LMN

3.In the figure, 

YX−→− and YZ−→− are tangents to circle A at points X and Z, respectively.  

mXZ=140∘
mXSZ=220∘

 What is the measure of  ∠XYZ ?

Enter your answer in the box.

4.AC

←→ is tangent to the circle with center at B. The measure of  ∠ACB is 58°.

What is the measure of  ∠ABC ?

 Enter your answer in the box.

m∠ABC = °

1This figure shows circle O with chords AC and BD mAB34 mCD34 AP6 m PC8 m What is BD Enter your answer in the box2This figure shows circle C with diameter LM an class=
1This figure shows circle O with chords AC and BD mAB34 mCD34 AP6 m PC8 m What is BD Enter your answer in the box2This figure shows circle C with diameter LM an class=
1This figure shows circle O with chords AC and BD mAB34 mCD34 AP6 m PC8 m What is BD Enter your answer in the box2This figure shows circle C with diameter LM an class=
1This figure shows circle O with chords AC and BD mAB34 mCD34 AP6 m PC8 m What is BD Enter your answer in the box2This figure shows circle C with diameter LM an class=

Respuesta :

MrEquation
Answer: You have a lot of work in this one question. I will help you with the first part. The measure of line segment BD would be 14.

Given the information provided, we could say that congruent triangles could be formed with triangles ABP and triangle DCP.

Therefore, the distance of BD is equal to the distance of AC. AC is found by just putting together the 6 and 8 to get 14. So BD is also 14.

The measure of the angles formed by the chords and tangent are given by

circle theorems.

Correct responses:

  • 1. BD = 14 m
  • 2. [tex]m\widehat{LMN}[/tex] = 296°
  • 3. ∠XYZ = 40°
  • 4. m∠ABC = 32°

Method used in arriving at the above values;

1. Given

Chords of circle O = AC and BD

The measure of arc [tex]m \widehat{AB}[/tex] = 34°

The measure of arc [tex]\mathbf{m \widehat {CD}}[/tex] = 34°

AP = 6 m

PC = 8 m

Required:

The length of BD

Solution:

[tex]\mathbf{m \widehat{AB}}[/tex] = [tex]m \widehat {CD}[/tex] (Given)

Segment AB = Segment CD (Chord subtended by equal arcs)

∠APB ≅ ∠CPD (vertical angles theorem)

∠BAC = ∠CDB (angles subtended by same chord)

ΔAPB ≅ ΔCPD by Side-Angle-Angle SAA similarity postulate

AP ≅ DP by Corresponding Parts of Congruent Triangles are Congruent, CPCTC

PB ≅ PB by CPCTC

Therefore;

AP = DP = 6 m by definition of congruency

PB = PB = 8 m by definition of congruency

BD = PB + DP by segment addition property

Therefore;

  • BD = 6 m + 8 m = 14 m

2. Given:

∠LMN = 32°

Required:

The measure of arc [tex]\mathbf{m \widehat{LMN}}[/tex]

Solution:

[tex]m\widehat{LN}[/tex] = 2 × ∠LMN (angle subtended by an arc at center is twice angle at circumference)

[tex]m \widehat{LN}[/tex] = 2 × 32° = 64°

[tex]m \widehat{LMN}[/tex] + [tex]m\widehat{LN}[/tex] = 360° (sum of angles at the center of a circle)

[tex]m \widehat{LMN}[/tex] = 360° - [tex]\mathbf{m\widehat{LN}}[/tex]

  • [tex]m \widehat{LMN}[/tex]= 360° - 64° = 296°

3. Given;

[tex]m \widehat{XZ}[/tex] = 140°

[tex]m \widehat{XSZ}[/tex] = 220°

Required:

The measure of ∠XYZ

Solution:

The angle ∠XYZ formed outside the circle by the two tangents [tex]\overrightarrow{YX}[/tex] and [tex]\overrightarrow{YZ}[/tex] is given by the formula;

[tex]\displaystyle \angle XYZ = \mathbf{\frac{m\widehat{XSZ}- m\widehat{XZ}}{2}}[/tex]

Therefore;

[tex]\displaystyle \angle XYZ = \frac{220^{\circ} - 140^{\circ}}{2} = \mathbf{ 40^{\circ}}[/tex]

  • ∠XYZ = 40°

4. Given:

[tex]\overleftrightarrow{AC}[/tex] is tangent to the circle

∠ACB = 58°

Required:

The measure of ∠ABC

Solution:

∠BAC = 90° (definition of a tangent of a circle [tex]\mathbf{\overleftrightarrow{AC}}[/tex])

∠ACB + ∠BAC + ∠ABC = 180° (angle sum property of a triangle)

∠ABC = 180° - (∠ACB + ∠BAC)

  • ∠ABC = 180° - (58 + 90°) = 32°

Learn more about chords and tangents of a circle here:

https://brainly.com/question/1423054

https://brainly.com/question/22413945