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As shown in the figure, ad is the diameter of ⊙ o, AC is the chord, ∠ CAD = 30 °, ob ⊥ ad is in O, AC is in B, ab = 5, and the length of BC is obtained
Connect the CD,
∵ ad is the diameter of ⊙ o,
∴∠C=90°，
∵OB⊥AD，
∴∠AOB=∠C=90°，
In RT △ AOB,
∵∠CAD=30°，AB=5，
∴OB=5
2，OA=OB•cot30°=5
2 x
3=5
Three
2，
∴AD=5
3，
∴AC=AD•cos30°=5
3 x
Three
2=15
2，
∴BC=AC-AB=15
2-5=5
2．
As shown in figure a, B, C, D, the point is on the circle O, ad is the diameter of circle O, ad = 6cm, if ∠ ABC = ∠ CAD, calculate the chord AC length
Magic wand,
From ∠ ABC = ∠ CAD
Chord AC = chord CD
therefore
String ad = chord AC + chord CD = 3.14x6 = 18.84cm
Chord AC = 18.84 △ 2 = 9.42cm
As shown in the figure, OC is the angular bisector of AOB, P is a point on OC. PD ⊥ OA intersects OA in D, PE ⊥ ob intersects ob in E, f is another point on OC, connecting DF and ef
It is proved that: ∵ point P is on the angular bisector OC of ∠ AOB, PE ⊥ ob, PD ⊥ Ao, PD = PE, ∵ DOP = ∠ EOP, ∠ PDO = ∠ PEO = 90 °, DPF = 90 ° - ∠ EOP, ? DPF = ∠ EPF, (2 points) in △ DPF and △ EPF, PD = PE ∠ DPF = PF (SAS)
In circle O, we know that the diameter ab of circle O is 2, the length of chord AC is root three, and the length of chord ad is root two. What is the square of DC?
If the length of chord AC is the root sign, the angle ∠ Cao = 30 degrees is obtained from the vertical diameter theorem. Similarly, the angle Dao = 45 ° cos (∠ CaO + ∠ Dao) = cos (30 + 45) = (√ 6) / 4 - (√ 2) / 4 cosine theorem cos (∠ CaO + ∠ Dao) = (AC ^ 2 + ad ^ 2-DC ^ 2) / (2Ac * ad) = (3 + 2-DC ^ 2) / 2 √ 6 is equal, so DC ^ 2 = 2 + √ 3
In circle O, if the diameter ab of circle O is known to be 2, the length of string AC is root 3, and the length of string ad is root 2, then the square of DC is equal to the help of the gods
So, if we make the center point of CE = 1, then we can get the value of edce = 1 + 2
In circle O, if the diameter of circle O is known to be 2, the chord AC is root 3, and chord ad is root 2, then DC ^ 2 = detailed steps
Let the diameter be ab,
1. If C and D are on both sides of ab
∵ ACB = 90 degrees
∴sin∠ABC=AC/AB=√3/2
Ψ ADC = ∠ ABC = 60 degrees
∵ ADC = 90 degrees
∴sin∠ABD=AD/AB=√2/2
Ψ ACD = ∠ abd = 45 degrees
﹤ CAD = 180 degrees - ﹤ ACD - ∠ ADC = 75 degrees
from the cosine theorem CD AC 2 AC cos is 75 degrees
=3+2-2*√3*√2*(√6-√2)/4
=2+√3
2. If C and D are on the same side of ab
Then ∠ CAD = ∠ bad - ∠ BAC = 15 degrees
According to the cosine theorem CD 2 = AC 2 + Ad 2 - 2 ad * ac * cos 15 degrees
=3+2-2*√3*√2*(√6+√2)/4
=2-√3
In circle O, if the diameter of circle O is known as ab = 2, the length of chord AC is root 3, and the length of chord ad is root 2, then there are two cases of square = CD
Connect BC, BD, OC, OD,
∵ diameter AB = 2, chord AC = √ 3, chord ad = √ 2,
∴∠CAB＝30º,∠DAB＝45º,
∴∠COB＝60º,∠DOB＝90º；
(1) when AC and AD are on the same side of AB, ∠ cod = 90 ° - 60 ° = 30 °,
In △ cod, it is obtained from cosine theorem
CD²＝1²＋1²－2×1×1×cos30º
＝2－√3,
(2) when AC and AD are on the opposite side of AB, ∠ cod = 90 ° C + 60 ° C = 150 ° C,
In △ cod, it is obtained from cosine theorem
CD²＝1²＋1²－2×1×1×cos150º
＝2＋√3.
AB is the diameter of circle O, AC and AD are the two chords of circle O. given AB = 16, AC = 8, ad = 8, find the angle of ∠ DAC
∵ AB is the diameter, ACB = ∠ ADB = 90 °,
cos∠CAB=AC/AB=1/2,∴∠CAB=60°,
∵ AC = ad = 8, ᙽ C and D are on the opposite side of AB respectively,
∴∠CAD=120°.
In the circle O, diameter AB = 2, chord AC = radical 3, chord ad = radical 2, find CD
Connect BC, BD
AB is the diameter, so ∠ ACB = ∠ ADB = 90
In right triangle ABC, Pythagorean theorem, BC=1
Because AB = 2, so ∠ BAC = 30 degrees
Similarly, BD = ad = √ 2
So ∠ bad = 45 degrees
∠ CAD = ∠ BAC + ∠ bad = 75 degrees
∠ ADC = ∠ ABC = 90 - ∠ BAC = 90-30 = 60 degrees
Sine theorem
AC/sin∠ADC=CD/sin∠DAC
√3/sin60=CD/sin75
√3/(√3/2)=CD/[(√6+√2)/4]
CD=(√6+√2)/2
CD²=(8+4√3)/4=2+√3
Given that the diameter ab of circle O is 2, the length of chord AC is root 3, and the length of chord ad is root 2. What is DCI ^ 2? It doesn't matter if there's no diagram. Write the process clearly,
The above solution ignores another position relation
When C and D are on the same side of diameter AB, CD * CD = 2 - √ 3
When C and D are on the opposite side of diameter AB, CD * CD = 2 + √ 3
The process has just been written out. I don't know why it is always prompted and not suitable to publish characters when submitting
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