Geometry Mathematics 2 Set 1 2021-2022 SSC (English Medium) 10th Standard Board Exam Question Paper Solution

If ΔABC ∼ ΔDEF and ∠A = 48°, then ∠D = ______.

48°

83°

49°

132°

AP is a tangent at A drawn to the circle with centre O from an external point P. OP = 12 cm and ∠OPA = 30°, then the radius of a circle is ______.

12 cm

`6sqrt(3)` cm

6 cm

`12sqrt(3)` cm

Seg AB is parallel to X-axis and coordinates of the point A are (1, 3), then the coordinates of the point B can be ______.

(–3, 1)

(5, 1)

(3, 0)

(–5, 3)

The value of 2tan45° – 2sin30° is ______.

2

1

`1/2`

`3/4`

In ΔABC, ∠ABC = 90°, ∠BAC = ∠BCA = 45°. If AC = `9sqrt(2)`, then find the value of AB.

Chord AB and chord CD of a circle with centre 0 are congruent. If m(arc AB) = 120°, then find the m(arc CD).

Find the Y-coordinate of the centroid of a triangle whose vertices are (4, –3), (7, 5) and (–2, 1).

If sinθ = cosθ, then what will be the measure of angle θ?

In the above figure, seg AC and seg BD intersect each other in point P. If `("AP")/("CP") = ("BP")/("DP")`, then complete the following activity to prove ΔABP ∼ ΔCDP.

Activity: In ΔABP and ΔCDP

`("AP")/("CP") = ("BP")/("DP")` ......`square`

∴ ∠APB ≅ `square` ......Vertically opposite angles

∴ `square` ∼ ΔCDP  ....... `square` test of similarity.

In the above figure `square`ABCD is a rectangle. If AB = 5, AC = 13, then complete the following activity to find BC.

Activity: ΔABC is a `square` triangle.

∴ By Pythagoras theorem

AB² + BC² = AC²

∴ 25 + BC² = `square`

∴ BC² = `square`

∴ BC = `square`

Complete the following activity to prove:

cotθ + tanθ = cosecθ × secθ

Activity: L.H.S. = cotθ + tanθ

= `cosθ/sinθ + square/cosθ`

= `(square + sin^2theta)/(sinθ xx cosθ)`

= `1/(sinθ xx  cosθ)` ....... ∵ `square`

= `1/sinθ xx 1/cosθ`

= `square xx secθ`

∴ L.H.S. = R.H.S.

If ΔABC ~ ΔPQR, AB : PQ = 4 : 5 and A(ΔPQR) = 125 cm², then find A(ΔABC).

In the above figure, m(arc DXE) = 105°, m(Arc AYC) = 47°, then find the measure of ∠DBE.

Draw a circle of radius 3.2 cm and centre 'O'. Take any point P on it. Draw tangent to the circle through point P using the centre of the circle.

If sinθ = `11/61`, then find the value of cosθ using the trigonometric identity.

In ΔABC, AB = 9 cm, BC = 40 cm, AC = 41 cm. State whether ΔABC is a right-angled triangle or not. Write reason.

In the above figure, chord PQ and chord RS intersect each other at point T. If ∠STQ = 58° and ∠PSR = 24°, then complete the following activity to verify:

∠STQ = `1/2` [m(arc PR) + m(arc SQ)]

Activity: In ΔPTS,

∠SPQ = ∠STQ – `square`  ......[∵ Exterior angle theorem]

∴ ∠SPQ = 34°

∴ m(arc QS) = 2 × `square`° = 68°   ....... ∵ `square`

Similarly, m(arc PR) = 2∠PSR = `square`°

∴ `1/2` [m(arc QS) + m(arc PR)] = `1/2` × `square`° = 58°  ......(I)

But ∠STQ = 58°  .....(II) (given)

∴  `1/2` [m(arc PR) + m(arc QS)] = ∠______  ......[From (I) and (II)]

Complete the following activity to find the coordinates of point P which divides seg AB in the ratio 3:1 where A(4, – 3) and B(8, 5).

Activity:

∴ By section formula,

∴ x = `("m"x_2 + "n"x_1)/square`, 

∴ x = `(3 xx 8 + 1 xx 4)/(3 + 1)`,

= `(square + 4)/4`,

∴ x = `square`,

∴ y = `square/("m" + "n")`

∴ y = `(3 xx 5 + 1 xx (-3))/(3 + 1)`

= `(square - 3)/4`

∴ y = `square`

In ΔABC, seg XY || side AC. If 2AX = 3BX and XY = 9, then find the value of AC.

Prove that "Opposite angles of a cyclic quadrilateral are supplementary".

ΔABC ~ ΔPQR. In ΔABC, AB = 5.4 cm, BC = 4.2 cm, AC = 6.0 cm, AB:PQ = 3:2, then construct ΔABC and ΔPQR.

Show that: `tan "A"/(1 + tan^2 "A")^2 + cot "A"/(1 + cot^2 "A")^2 = sin"A" xx cos"A"`

□ABCD is a parallelogram. Point P is the midpoint of side CD. seg BP intersects diagonal AC at point X, then prove that: 3AX = 2AC

In the above figure, seg AB and seg AD are tangent segments drawn to a circle with centre C from exterior point A, then prove that: ∠A = `1/2` [m(arc BYD) - m(arc BXD)]

Find the co-ordinates of centroid of a triangle if points D(–7, 6), E(8, 5) and F(2, –2) are the mid-points of the sides of that triangle.

If a and b are natural numbers and a > b If (a² + b²), (a² – b²) and 2ab are the sides of the triangle, then prove that the triangle is right-angled. Find out two Pythagorean triplets by taking suitable values of a and b.

Construct two concentric circles with centre O with radii 3 cm and 5 cm. Construct a tangent to a smaller circle from any point A on the larger circle. Measure and write the length of the tangent segment. Calculate the length of the tangent segment using Pythagoras' theorem.