Based on the diagram below, what is the measure of arc DC ?
A 28 degrees
B 68 degrees
C 108 degrees
D 112 degrees
Answers
Given:
A diagram of a circle O, [tex]arc(AB)=y^\circ, arc(CD)=(80+y)^\circ[/tex].
To find:
The measure of arc DC.
Solution:
According to the angle of intersecting chords theorem:
Angle at intersection of chords = Half of the sum of intercepted arcs.
Let [tex]\theta [/tex] be the angle between the intersection of chords AC and BD.
[tex]\theta +112^\circ=180^\circ[/tex] (Linear pair)
[tex]\theta =180^\circ-112^\circ[/tex]
[tex]\theta =68^\circ[/tex]
Using the angle of intersecting chords theorem, we get
[tex]\theta=\dfrac{arc(AB)+arc(CD)}{2}[/tex]
[tex]68^\circ =\dfrac{y^\circ+(80+y)^\circ}{2}[/tex]
[tex]136^\circ =(80+2y)^\circ[/tex]
On simplification, we get
[tex]136=80+2y[/tex]
[tex]136-80=2y[/tex]
[tex]56=2y[/tex]
Divide both sides by 2.
[tex]28=y[/tex]
Now,
[tex]arc(DC)=(80+y)^\circ[/tex]
[tex]arc(DC)=(80+28)^\circ[/tex]
[tex]arc(DC)=108^\circ[/tex]
Therefore, the correct option is C.
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AD and MN are chords that intersect at point B.
A circle is shown. Chords A D and M N intersect at point G. The length of A B is 9, the length of B D is x + 1, the length of M B is x minus 1, and the length of B N is 15.
What is the length of line segment MN?
Answers
Answer:
[tex]MN = 18[/tex]
Step-by-step explanation:
Given
[tex]AB = 9[/tex]
[tex]BD = x + 1[/tex]
[tex]MB = x - 1[/tex]
[tex]BN = 15[/tex]
See attachment
Required
Determine MN
The products of the segments of two chords that intersect are always, equal.
So, we have:
[tex]AB * BD = MB * BN[/tex]
[tex]9 * (x + 1) = (x - 1) * 15[/tex]
Open bracket
[tex]9x + 9 = 15x - 15[/tex]
Collect like terms
[tex]15x - 9x = 9 + 15[/tex]
[tex]6x = 24[/tex]
Solve for x
[tex]x = \frac{24}{6}[/tex]
[tex]x = 4[/tex]
The length of MN is calculated as:
[tex]MN = MB + BN[/tex]
[tex]MN = x - 1 + 15[/tex]
Substitute [tex]x = 4[/tex]
[tex]MN = 4 - 1 + 15[/tex]
[tex]MN = 18[/tex]
Answer:
C. 18 units
Step-by-step explanation:
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Answer and Step-by-step explanation:
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