SSC (English Medium) 10th Standard - Maharashtra State Board Question Bank Solutions for Geometry Mathematics 2

In right-angled ΔABC, BD ⊥ AC. If AD= 4, DC= 9, then find BD.

The perpendicular height of a cone is 12 cm and its slant height is 13 cm. Find the radius of the base of the cone. 

A circle is inscribed in square ABCD of side 14 cm. Complete the following activity to find the area of the shaded portion.
  Activity:

Area of square ABCD = ______

= 14²  
 = 196 cm² 
Area of circle = πr²     = `22/7xx 7^2`   
= ____ cm²

Area of shaded portion = Area of square ABCD – Area of circle   

 = 196 – _______

= _____ cm²

In the given figure, M is the centre of the circle and seg KL is a tangent segment. L is a point of contact. If MK = 12, KL = `6sqrt3`, then find the radius of the circle.

Prove the following theorem:

Tangent segments drawn from an external point to the circle are congruent.

In the given figure `square`ABCD is a square of side 50 m. Points P, Q, R, S are midpoints of side AB, side BC, side CD, side AD respectively. Find area of shaded region

As shown in figure, ∠DFE = 90°, FG ⊥ ED, If GD = 8, FG = 12, then (1) EG = ? (2) FD = ? (3) EF = ?

Segment DP and segment DQ are tangent segments to the circle with center A. If DP = 7 cm. So find the length of the segment DQ?

The chords corresponding to congruent arcs of a circle are congruent. Prove the theorem by completing following activity.

Given: In a circle with centre B 

arc APC ≅ arc DQE

To Prove: Chord AC ≅ chord DE

Proof: In ΔABC and ΔDBE,

side AB ≅ side DB   ...`square`

side BC ≅ side `square`   ...`square`

∠ABC ≅ ∠DBE   ...[Measure of congruent arcs]

∆ABC ≅ ∆DBE   ...`square`

Length of a tangent segment drawn from a point which is at a distance 15 cm from the centre of a circle is 12 cm, find the diameter of the circle?

Tangent segments drawn from an external point to a circle are congruent, prove this theorem. Complete the following activity.

Given: `square`

To Prove: `square`

Proof: Draw radius AP and radius AQ and complete the following proof of the theorem.

In ∆PAD and ∆QAD,

seg PA ≅ `square`   ...[Radii of the same circle]

seg AD ≅ seg AD   ...[`square`]

∠APD ≅ ∠AQD = 90°   ...[Tangent theorem]

∴ ∆PAD ≅ ∆QAD   ...[`square`]

∴ seg DP ≅ seg DQ   ...[`square`]

In the adjoining figure, O is the center of the circle. From point R, seg RM and seg RN are tangent segments touching the circle at M and N. If (OR) = 10 cm and radius of the circle = 5 cm, then

1. What is the length of each tangent segment?
2. What is the measure of ∠MRO?
3. What is the measure of ∠MRN?

In the adjoining figure circle with Centre, Q touches the sides of ∠MPN at M and N. If ∠MPN = 40°, find measure of ∠MQN.

The figure ΔABC is an isosceles triangle with a perimeter of 44 cm. The sides AB and BC are congruent and the length of the base AC is 12 cm. If a circle touches all three sides as shown in the figure, then find the length of the tangent segment drawn to the circle from the point B.

If AB and CD are the common tangents in the circles of two unequal (different) radii, then show that seg AB ≅ seg CD.

Seg RM and seg RN are tangent segments of a circle with centre O. Prove that seg OR bisects ∠MRN as well as ∠MON with the help of activity.

Proof: In ∆RMO and ∆RNO,

∠RMO ≅ ∠RNO = 90°   ...[`square`]

hypt OR ≅ hypt OR   ...[`square`]

seg OM ≅ seg `square`   ...[Radii of the same circle]

∴ ∆RMO ≅ ∆RNO   ...[`square`]

∠MOR ≅ ∠NOR

Similairy ∠MRO ≅ `square`   ...[`square`]

Prove that, tangent segments drawn from an external point to the circle are congruent.

If x = `θ/360` × 2πr then what is x in the formula?

In the given figure, a rectangle ABCD is inscribed inside a semi-circle of radius 10 cm. Using the dimensions given in the figure, determine the area of the shaded region.

In a parallelogram ABCD, ∠B = 105°. Determine the measure of ∠A and ∠D.