10th Class Mathematics Unit 9 Tangent and Angles Numericals with Solutions

9.1.1.Find the value of x. (In the figure, O is the centre of the circle, A is the point of tangency, B is a point on the tangent line, ∠ABO = 38°, and x = ∠AOB.)

Given

Given angle angle ABO = 38^circ

Formula

Tangent–radius theorem OA perp tangent at A ⇒ angle OAB = 90^circ

Angle sum of \triangle AOB angle AOB + angle OAB + angle ABO = 180^circ

Working

Substituting x + 90^circ + 38^circ = 180^circ

x + 128^circ = 180^circ

Result

Value of x x = 52^circ

9.1.2.Find the angle ABC. (AB and AC are tangents to a circle with centre O, touching it at B and C, and ∠A = 65°.)

Given

Given angle angle A = 65^circ

Formula

Tangents from external point AB = AC ⇒ triangle ABC is isosceles, angle B = angle C

Angle sum of \triangle ABC angle A + angle B + angle C = 180^circ

Working

Substituting 65^circ + 2angle B = 180^circ

2angle B = 115^circ ⇒ angle B = 115^circ2

Result

Angle ABC angle ABC = 57.5^circ

9.1.3.A, B and C are points on the circumference of a circle with centre at O. AC is the diameter of the circle and DE is the tangent to the circle at the point C, and m∠BCD = 62°. Find (i) m∠BCA (ii) m∠BAC.

Given

Given angle angle BCD = 62^circ

Formula

Alternate segment theorem angle BCD = angle BAC = 62^circ

Angle in a semicircle AC is diameter ⇒ angle ABC = 90^circ

Angle sum of \triangle ABC angle BCA + angle ABC + angle BAC = 180^circ

Working

Substituting angle BCA + 90^circ + 62^circ = 180^circ

Result

Answers angle BCA = 28^circ, angle BAC = 62^circ

9.1.4.D, E and F are points on the circumference of a circle with centre O and m∠DOF = 130°. Find m∠DEF.

Given

Given angle angle DOF = 130^circ

Formula

Central angle theorem mangle DEF = 12 × mangle DOF

Working

Substituting = 12 × 130^circ

Result

Angle DEF mangle DEF = 65^circ

9.1.5.X, Y and Z are points on the circumference of a circle with centre O and m∠XYZ = 85°. Find x. (x is the central angle subtending the same arc XZ.)

Given

Given angle angle XYZ = 85^circ

Formula

Central angle theorem x = 2 × angle XYZ

Working

Substituting x = 2 × 85^circ

Result

Value of x x = 170^circ

9.1.6.In a historical monument, a circular fountain with a radius of 3 m is built. A flagpole is erected 7 m away from the centre of the fountain. Two ropes from the pole are tied to the edge of the fountain, just touching it. Find the length of each rope.

Given

Given OT = 3 m, OP = 7 m

Formula

Pythagoras theorem in right \triangle OTP PT² = OP² - OT²

Working

Substituting PT² = 7² - 3² = 49 - 9 = 40

PT = sqrt{40} = sqrt{4 × 10} = 2sqrt{10} m

Result

Length of each rope 2sqrt{10} m

9.1.7.Two circular gears touch each other externally for proper rotation in a machine. The radius of the two circular gears are 5 cm and 7 cm. What is the distance between their centres if they touch externally?

Given

Radii r₁ = 5 cm, r₂ = 7 cm

Formula

Circles touching externally Distance between centres = Sum of radii

Working

Substituting = 5 cm + 7 cm

Result

Distance between centres 12 cm

9.1.8.A small sensor lies inside a satellite dish and touches its wall internally. If the dish has radius 15 cm and the sensor has radius 2.5 cm, find the distance between their centres.

Given

Radii R = 15 cm, r = 2.5 cm

Formula

Circles touching internally Distance between centres = Difference of radii

Working

Substituting = 15 cm - 2.5 cm

Result

Distance between centres 12.5 cm

9.1.9.An inner holder touches the outer cylindrical container internally. If the outer container has radius 8 cm and the inner holder has radius 6 cm, find the distance between their centres.

Given

Radii R = 8 cm, r = 6 cm

Formula

Circles touching internally Distance between centres = Difference of radii

Working

Substituting = 8 cm - 6 cm

Result

Distance between centres 2 cm

9.1.10.A pyramid-shaped sculpture is placed in the center of a circular plaza with a radius of 10 m. A decorative pole stands 26 m from the center of the circle. Two guide wires are attached from the pole to the plaza edge, just touching the circle. Find the length of each wire.

Given

Given OT = 10 m, OP = 26 m

Formula

Pythagoras theorem in right \triangle OTP PT² = OP² - OT²

Working

Substituting PT² = 26² - 10² = 676 - 100 = 576

PT = sqrt{576} = 24 m

Result

Length of each wire 24 m

Unit 9: Tangent and Angles — Exercise 9 1