Geometrical Proofs

QUESTIONS ON GEOMETRICAL PROOFS − 10

D

C A B In the figure above, the line AB is a tangent and the line ACD is a secant to the circle respectively. Prove that if BD is a diameter of the circle, then BC 2 = AC × CD . CD Given that sin ∠BAC = 0.8 , find the exact value of . [7] AD

8. A

34 

T D

56 

P

B

C S

In the diagram above SCT is a tangent to a circle at the point C. The points A, B and D are points that lie on the circle such that AC and BD intersect at the point P. Given that AB is parallel to CD , ∠CAD = 34  and ∠ABD = 56  . (a) Find ∠BTC . (b) Show that AC is a diameter of the circle. (c) Give a reason why CT 2 = AT 2 − AC 2 . (d) Prove that BD × DT = AT 2 − AB 2 − BC 2 − DT 2 .

[4] [3] [2] [5]

In the figure, ∠ BAC = ∠ AEC = 90 o . CF = CA. Prove that ∠ BAF = ∠ FAE. B F

A

E

C

[3]

6

In the diagram, the tangent at P meets XY produced at Z. (i)

Show that ∆ PYZ is similar to ∆ XPZ.

[3]

(ii)

Prove that PZ × PY = PX × YZ .

[1]

X

(iii)

Y 2

PX XZ Hence show that . = 2 YZ PY

P

Z

[3]

11

In, the figure, the circle intersects the triangle ABC at P, D, F, C and E. Given that PF is parallel to BC and AT is a tangent to the circle at T , prove that

FC PB = AD AC

[5]

A

T

D

P F

B

E

C

11. In the diagram, a circle, C1 with centre O passes through the vertices of the triangle ABC. The diameter AC is produced to E such that the perpendicular from AB through C meets BC produced at D. AT is a tangent to the circle C1 and angle BFA is a right angle. CB produced and AF produced meet at T. T B

E

F

C

O

A

D

(a) Name two triangles which are similar to ∆ABC, showing clearly your reasons. Hence show that (i) BC × DC = AC × EC 2 (ii) AB = AC × BF (b) (b)

[4] [1] [1]

Show that TB 2 = TC × TB − AC × BF Another circle, C2 passes through the points A, B, D and E. State the centre of this circle, showing your reasons clearly.

[2] [2]

11.

F

P

D

C

A B E In the diagram above, P is any pointQon theOsemi-circle with centre O and PQ is perpendicular to AB. The inscribed circle with centre C touches PQ, AB and the semicircle at D, E and F respectively such that DC is parallel to AO. Prove that (i) ∆ADQ is similar to ∆ABF, (ii) AD × AF = AQ × AB, (iii) AE2 = AQ2 + AQ × QB, (iv) AD × CF = OC × DF.

[3] [1] [4] [2]

7 A B E H C

D

F

G In the circle, B, E, C, F and D are points on the circle such that CD is a diameter. The straight lines ABC and FE intersect at the point H. ADG is a tangent to the circle at D. (a)

Show that CD2 = AC.BC

(b)

If ∠BDE = ∠FDG, prove that BD is parallel to EF.

[2]

(c)

State, with reasons, why angle BHF = 90o.

[2]

[4]

11

The diagram belows shows a circle with centre P. AD is the

diameter. DC is a tangent to the circle and ∠BCD is a right angle. AB produced and DC produced meet at T.

A

B P

D

(a)

(b)

C

T

Show that ∆ABD and ∆DCB are similar.

[3]

Name three other triangles which are similar to ∆ABD.

[3]

Giving the reasons clearly, show that (i)

DB 2 = DA × CB ,

(ii) DC 2 = CB × ( DA − CB ) .

[2] [2]

(b)

In the diagram ADB is a tangent to a circle at point D. The points E, C and F lie on the circle. AFC and BEC are straight lines and the chords EC and DF are parallel. C

E F

A

D

B

(i) (ii)

AF . FC Hence prove that AD × DB = AC × FC . Prove that AD = DB ×

[2] [3]

9.

In the diagram, PQ is a common chord of the two circles PQR and PQBC. QC is a tangent to the circle PQR at Q. QC and BP intersect at S. (i)

Prove that BC is parallel to QR.

R

P S

[3]

(ii) Prove that ∆PQS is similar to ∆QRS . Hence, show that QS2 = PS × RS. [3]

C

Q

B