ABC is an inscribed triangle and chord AD bisects angle A and cuts BC at P. Prove: (AD)/(CD)=(AC)/(PC).

GIven: Inscribed △ ABC, chord AD bisects ∠ A and cuts BC at P; chord CD

Prove: (AD)/(CD)=(AC)/(PC).

Statements

1.(see above)

2. ∠ D=∠ D

3.

4. ∠ BCD=∠ DAB

5.

6. △ ACD-△ CPD

7. (AD)/(CD)=(AC)/(PC)

Reasons

1.Given

2.

3.Defintion of angle bisectors

4.

5.Substitution

6.

7.

GIven: Inscribed △ ABC, chord AD bisects ∠ A and cuts BC at P; chord CD

Prove: (AD)/(CD)=(AC)/(PC).

Statements

1.(see above)

2. ∠ D=∠ D

3.

4. ∠ BCD=∠ DAB

5.

6. △ ACD-△ CPD

7. (AD)/(CD)=(AC)/(PC)

Reasons

1.Given

2.

3.Defintion of angle bisectors

4.

5.Substitution

6.

7.

1 Answers

Best Answer

**Step1**∠D=∠ D Here angle D is equal to itself, that is reflexive property. Given, AD bisects angle A, so by definition of angle bisector ∠ CAD=∠ DAB

**Step2**From the picture, we see that angle BCD and angle DAB are two inscribed angles with the same base points on the circle. So they are equal. ∠ BCD=∠DAB If we substitute angle DAB by angle BCD in ∠ CAD=∠ DAB We get: ∠ BCD=∠ CAD So triangle ACD and triangle CPD has two pairs of angles equal. So they are ~ilar by AA property. Corollary 57-1 If two angles of one triangle are equal respectively to two angles of another, then the triangles are ~ilar. (a.a.) So, (AD)/(CD)=(AC)/(PC) by C.S.S.T.P, - corresponding sides of ~ilar triangles are proportional. Answer: 2) Reflexive property 3) ∠ CAD=∠ DAB 4)Inscribed angles with same base points 5) ∠ BCD=∠ CAD 6) a.a 7) C.S.S.T.P.