10th Class Mathematics Unit 8 Chords and Arcs Numericals with Solutions

8.2.1.In the given figure, if PQ = QR = RS = ST and m∠POT = 70°, then find the value of x (the central angle subtended by each of the equal arcs).

Given

Given widehat{PQ} = widehat{QR} = widehat{RS} = widehat{ST}

The corresponding arcs are equal ⇒ central angles subtending them are equal. Let each = x.

Formula

Angles around a point 70^circ + x + x + x + x = 360^circ

Working

Substituting 70^circ + 4x = 360^circ

4x = 360^circ - 70^circ = 290^circ ⇒ x = 290^circ4

Result

Value of x x = 72.5^circ

8.2.2.In the given figure, find the values of x and y. (Given: the central angle is 154°, x is the inscribed angle subtending the same arc, arc AB = 3 cm, and AB = BC = CD = AD.)

Given

Property The angle at the centre is twice the angle at the circumference subtending the same arc.

Working

Substituting x = 12 × 154^circ

Result

Value of x x = 77^circ

Given

Equal chords AB = BC = CD = AD

Equal chords subtend equal arcs ⇒ Arcs AB = BC = CD = DA

Working

Substituting Arc AB = 3 cm ⇒ Arc DC = y = 3 cm

Result

Value of y y = 3 cm

8.2.3.Find the values of x and y in the given figure, such that mAB = mBD. (Given: central angle AOD = 80°, central angle AOC = 52°, with x = angle COD, arc AB = 6 cm.)

Given

Given mwidehat{AB} = mwidehat{BD}

Central angles subtending these arcs are equal ⇒ angle AOB = angle BOD

Working

Substituting 80^circ = 52^circ + x

Result

Value of x x = 80^circ - 52^circ = 28^circ

Given

Equal arcs Equal arcs have equal lengths, so overline{AB} = overline{BD}

AB = BD

Working

Substituting 6 cm = y cm

Result

Value of y y = 6 cm

8.2.4.Two congruent arcs in a circular track subtend angles of 60° each at the centre. If the length of one chord of the circle is 10 metres, what is the length of other chord?

Given

Given Congruent arcs subtend equal central angles; both arcs subtend 60^circ.

Formula

Property Chords subtending equal central angles are equal.

Result

Length of other chord Length of other chord = Length of given chord = 10 metres

8.2.5.In a circular fountain, two water jets are installed such that they spray water along arcs of equal length. If one jet sprays between points P and Q and the straight-line distance (chord PQ) is 12 metres, what is the straight-line distance (chord RS) covered by the second jet spraying along arc RS, which is congruent to arc PQ?

Given

Given Arc PQ cong Arc RS (equal arcs), quad Chord PQ = 12 m

Formula

Property In the same (or congruent) circle, congruent arcs subtend congruent chords.

Result

Distance (chord RS) Chord RS = Chord PQ = 12 metres

8.2.6.In a circular park, two walkways AB and CD are both straight paths (chords) of length 14 metres. What can be said about the minor arcs subtended by these walkways?

Given

Given Chord overline{AB} = 14 m, Chord overline{CD} = 14 m

Formula

Property In the same circle, equal chords subtend equal arcs.

Result

Conclusion The minor arcs subtended by overline{AB} and overline{CD} are congruent.

8.2.7.In two congruent circular clocks, the minute hand points from the centre to the 3 on both. A decorative string connects 3 to 9 on both clocks. Are the arcs from 3 to 9 on both clocks congruent?

Given

Given Both clocks are congruent circles.

From 3 to 9 is a semicircle in each clock.

Formula

Central angle Central angle between 3 and 9 = 180^circ in both

Result

Conclusion Arcs from 3 to 9 on both clocks are congruent.

Unit 8: Chords and Arcs — Exercise 8 2