In a circle two chords AB & CD both subtend 50 at the centre O If AB 6cm then find CD

In a circle, two chords AB and CD intersect at a point P inside the circle. Prove that
(a) ΔPAC ∼PDB (b) PA. PB= PC.PD

Given : AB and CD are two chordsTo Prove:(a) Δ PAC ~ ΔPDB(b) PA.PB = PC.PDProof: In Δ PAC and Δ PDB∠𝐴𝑃𝐶 = ∠𝐷𝑃𝐵 (𝑉𝑒𝑟𝑡𝑖𝑐𝑎𝑙𝑙𝑦 𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑎𝑛𝑔𝑙𝑒𝑠)∠𝐶𝐴𝑃 = ∠𝐵𝐷𝑃 (𝐴𝑛𝑔𝑙𝑒𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑎𝑚𝑒 𝑠𝑒𝑔𝑚𝑒𝑛𝑡 𝑎𝑟𝑒 𝑒𝑞𝑢𝑎𝑙)𝑏𝑦 𝐴𝐴 𝑠𝑖𝑚𝑖𝑙𝑎𝑟𝑖𝑡𝑦 𝑐𝑟𝑖𝑡𝑒𝑟𝑖𝑜𝑛 Δ𝑃𝐴𝐶 ~𝑃𝐷𝐵

When two triangles are similar, then the ratios of lengths of their corresponding sides are proportional.

`∴ (PA)/(PD)=(PC)/(PB)`

⟹ PA. PB = PC. PD

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Page 2

Two chords AB and CD of a circle intersect at a point P outside the circle.
Prove that: (i) Δ PAC ~ Δ PDB (ii) PA. PB = PC.PD

Given : AB and CD are two chordsTo Prove:(a) Δ PAC - Δ PDB(b) PA. PB = PC.PDProof: ∠𝐴𝐵𝐷 + ∠𝐴𝐶𝐷 = 180° …(1) (Opposite angles of a cyclic quadrilateral aresupplementary)∠𝑃𝐶𝐴 + ∠𝐴𝐶𝐷 = 180° …(2) (Linear Pair Angles )Using (1) and (2), we get∠𝐴𝐵𝐷 = ∠𝑃𝐶𝐴

∠𝐴 = ∠𝐴 (Common)

By AA similarity-criterion Δ PAC - Δ PDB
When two triangles are similar, then the rations of the lengths of their corresponding sides are proportional

`∴ (PA)/(PD)=(PC)/(PB)`

⟹ PA.PB = PC.PD

Is there an error in this question or solution?

Question 3 Circles - Exercise 10.2

Answer:

False

Justification:

Let AB and AC be the chord of the circle with center O on the opposite side of OA.

Consider the triangles AOC and AOB:

AO = AO (Common side in both triangles)

OB = OC (Both OB and OC are radius of circle)

But we can’t show that either the third side of both triangles are equal or any angle is equal.

Therefore ΔAOB is not congruent to ΔAOC.

∴ ∠OAB ≠ ∠OAC.

Video transcript

"hello student welcome welcome back on a leader down solving session again and again today we come up with one one new problem of mathematics right basically this problem is related to the geometry right okay so quickly i will read the statement for you two chords a b and a c of the circles with center o are on the opposite side of wave then angle o a b is equals to angle of oac so basically we need to check whether this statement is correct or wrong right and to understand this what i have done i have drawn this figure okay you can see the figure here all right okay now quickly we will solve this problem right okay so from this figure okay we get two triangles here right so one triangle is triangle aoc and another triangle we get is triangle a ob all right okay so basically we are getting the two triangles here all right okay now let's discuss okay the sides of the triangle right so in this particular triangle okay side okay ao is equals to side okay ao right because both the triangles have the common side right so ao is a common side common side right okay that's why ao is equals to ao right okay now what we will do we will check okay again we can see that okay length of oc length of oc is equals to length of ob right okay because ob and oc are the radius of the circle right okay radius of the circle right so that's why okay ob and oc are the equivalent sides right okay they are having the equivalent okay but if you carefully observe that okay ac for the side ac and okay a b okay we can't say that whether these sides are equal or not right again we can't say that any of the angles are equal or not that's why we cannot predict the angle of a o a b and o a c or equal or not right and that's why angle at o a b okay is not equal to or it is not concurrent to and okay angle o a c okay so basically these are not the concurrent angle so that's all all about today's session please do comment if you have any doubt and please subscribe to this channel so that we will get the update for upcoming video bye you"

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