Question

Calculate the length of TQ​

Answer 1

x = 21°, y = 42°, TQ = 3.75 cm

Explanation:

In Δ SRT

∵ m∠RST = 21°, m∠STR = 117°

∵ The sum of the measures of the interior angles of a triangle is 180°

∴ m∠RST + m∠STR + m∠SRT = 180°

∴ 21° + 117° + m∠SRT = 180°

→ Add the like angles

∴ 138° + m∠SRT = 180°

→ Subtract 138° from both sides

∴ m∠SRT = 42°

In the given circle

∵ ∠SRP and ∠SQP are inscribed angles subtended by the arc SP

→ Inscribed angles subtended by the same arc are equal in measures

∴ m∠SRP = m∠SQP ⇒ (1)

∵ m∠SRP = 42° , m∠SQP = y

∴ y = 42°

∵ ∠RSQ and ∠RPQ are inscribed angles subtended by the arc RQ

→ Inscribed angles subtended by the same arc are equal in measures

∴ m∠RSQ = m∠RPQ ⇒ (2)

∵ m∠RSQ = 21° , m∠RPQ = x

∴ x = 21°

∵ The chord SQ intersects the chord PR at the point T

∴ ∠STR and ∠PTR are vertically opposite angles

→ The vertically opposite angles are equal in measures

∴ m∠STR = m∠PTR ⇒ (3)

→ From (1), (2), and (3) the two triangles SRT and PQT are similar using

AAA postulate of similarity

∴ Δ SRT ≈ ΔPQT ⇒ AAA postulate

∴ Their sides are proportion

∴
`(SR)/(PQ)=(RT)/(TQ)=(ST)/(PT)`

∵ SR = 8.32 cm, PQ = 9.43 cm, RT = 3.31 cm

→ Substitute them in the ratio above

∴
`(8.32)/(9.43)=(3.31)/(TQ)`

→ By using cross multiplication

∴ TQ × 8.32 = 9.43 × 3.31

→ Divide both sides by 8.32 to find TQ

∴ TQ =
`((9.43)(3.31))/(8.32)`

∴ TQ = 3.751598558

→ Round it to 2 d.p

∴ TQ = 3.75 cm