In the adjoining figure, M is the centre of the circle and seg AB is a

1.	In the adjoining figure, M is the centre of the circle and seg AB is a diameter, seg MS ⊥ chord AD, seg MT ⊥ chord AC, ∠DAB ≅ ∠CAB. i. Prove that: chord AD ≅ chord AC. ii. To solve this problem which theorems will you use? a. The chords which are equidistant from the centre are equal in length.b. Congruent chords of a circle are equidistant from the centre.iii. Which of the following tests of congruence of triangles will be useful?a. SAS b. ASA c. SSS d. AAS e. Hypotenuse-side test. Using appropriate test and theorem write the proof of the above example.
Answer» Proof: i. ∠DAB ≅ ∠CAB [Given] ∴ ∠SAM ≅ ∠TAM (i) [A – S – D, A – M – B, A -T – C] In ∆SAM and ∆TAM, ∠SAM ≅ ∠TAM [From (i)] ∠ASM ≅ ∠ATM [Each angle is of measure 90°] seg AM ≅ seg AM [Common side] ∴ ∆SAM ≅ ∆TAM [AAS [SAA] test of congruency] ∴ side MS ≅ side MT [c.s.c.t] But, seg MS ⊥ chord AD [Given] seg MT ⊥chord AC ∴ chord AD ≅ chord AC [Chords of a circle equidistant from the centre are congruent] ii. Theorem used for solving the problem: The chords which are equidistant from the centre are equal in length. iii. Test of congruency useful in solving the above problem is AAS ISAAI test of congruency.

In the adjoining figure, M is the centre of the circle and seg AB is a diameter, seg MS ⊥ chord AD, seg MT ⊥ chord AC, ∠DAB ≅ ∠CAB. i. Prove that: chord AD ≅ chord AC. ii. To solve this problem which theorems will you use? a. The chords which are equidistant from the centre are equal in length.b. Congruent chords of a circle are equidistant from the centre.iii. Which of the following tests of congruence of triangles will be useful?a. SAS b. ASA c. SSS d. AAS e. Hypotenuse-side test. Using appropriate test and theorem write the proof of the above example.

Answer»

Proof:

i. ∠DAB ≅ ∠CAB [Given]

∴ ∠SAM ≅ ∠TAM (i) [A – S – D, A – M – B, A -T – C]

In ∆SAM and ∆TAM,

∠SAM ≅ ∠TAM [From (i)]

∠ASM ≅ ∠ATM [Each angle is of measure 90°]

seg AM ≅ seg AM [Common side]

∴ ∆SAM ≅ ∆TAM [AAS [SAA] test of congruency]

∴ side MS ≅ side MT [c.s.c.t]

But, seg MS ⊥ chord AD [Given]

seg MT ⊥chord AC

∴ chord AD ≅ chord AC [Chords of a circle equidistant from the centre are congruent]

ii. Theorem used for solving the problem:

The chords which are equidistant from the centre are equal in length.

iii. Test of congruency useful in solving the above problem is AAS ISAAI test of congruency.