Geometry Mathematics 2 Model set 3 by shaalaa.com 2021-2022 SSC (English Medium) 10th Standard Board Exam Question Paper Solution

If the length of the segment joining point L(x, 7) and point M(1, 15) is 10 cm, then the value of x is ______

7

7 or – 5

–1

1

Choose the correct alternative:

If length of sides of a triangle are a, b, c and a² + b² = c², then which type of triangle it is?

Obtuse angled triangle

Acute angled triangle

Equilateral triangle

Right angled triangle

In fig, seg DE || sec BC, identify the correct statement.

`"AD"/"DB" = "AE"/"AC"`

`"AD"/"DB" = "AB"/"AC"`

`"AD"/"DB" = "EC"/"AC"`

`"AD"/"DB" = "AE"/"EC"`

Choose the correct alternative answer for the following question.

 1 cm

10 cm 

100 cm 

1000 cm

Choose the correct alternative:

∆ABC ∼ ∆AQR. `"AB"/"AQ" = 7/5`, then which of the following option is true?

A–Q–B

A–B–Q

A-C–B

A–R–B

The radius of a circle is 10 cm. The measure of an arc of the circle is 54^°. Find the area of the sector associated with the arc. (\[\pi\]= 3.14 )

ΔPQR ~ ΔSUV. Write pairs of congruent angles

Draw seg AB of length 4.5 cm and draw its perpendicular bisector

In figure, chord EF || chord GH. Prove that, chord EG ≅ chord FH. Fill in the blanks and write the proof.

Proof: Draw seg GF.

∠EFG = ∠FGH     ......`square`    .....(I)

∠EFG = `square`   ......[inscribed angle theorem] (II)

∠FGH = `square`   ......[inscribed angle theorem] (III)

∴ m(arc EG) = `square`  ......[By (I), (II), and (III)]

chord EG ≅ chord FH   ........[corresponding chords of congruent arcs]

From the given figure, in ∆ABC, if AD ⊥ BC, ∠C = 45°, AC = `8sqrt(2)` , BD = 5, then for finding value of AD and BC, complete the following activity.

Activity: In ∆ADC, if ∠ADC = 90°, ∠C = 45°    ......[Given]

∴ ∠DAC = `square`   .....[Remaining angle of ∆ADC]

By theorem of 45° – 45° – 90° triangle,

∴ `square = 1/sqrt(2)` AC and `square = 1/sqrt(2)` AC

∴ AD =`1/sqrt(2) xx square` and DC = `1/sqrt(2) xx 8sqrt(2)`

∴ AD = 8 and DC = 8

∴ BC = BD +DC

= 5 + 8

= 13

Find distance between point Q(3, – 7) and point R(3, 3)

Solution: Suppose Q(x₁, y₁) and point R(x₂, y₂)

x₁ = 3, y₁ = – 7 and x₂ = 3, y₂ = 3

Using distance formula,

d(Q, R) = `sqrt(square)`

∴ d(Q, R) = `sqrt(square - 100)`

∴ d(Q, R) =  `sqrt(square)`

∴ d(Q, R) = `square`

In the adjoining figure, circle with center D touches the sides of ∠ACB at A and B. If ∠ACB = 52°, find measure of ∠ADB.

In the given figure, if A(P-ABC) = 154 cm²  radius of the circle is 14 cm, find

(1) `∠APC` 

(2) l ( arc ABC) .

ΔABP ~ ΔDEF and A(ΔABP) : A(ΔDEF) = 144:81, then AB : DE = ?

Draw a circle of radius 3.4 cm, take any point P on it. Draw tangent to the circle from point P

In the given figure, m(arc NS) = 125°, m(arc EF) = 37°, find the measure ∠NMS.

From fig., seg PQ || side BC, AP = x + 3, PB = x – 3, AQ = x + 5, QC = x – 2, then complete the activity to find the value of x.

In ΔPQB, PQ || side BC

`"AP"/"PB" = "AQ"/(["______"])`    ...[______]

`(x + 3)/(x - 3) = (x + 5)/(["______"])`

(x + 3) [______] = (x + 5)(x – 3)

x² + x – [______] = x² + 2x – 15

x = [______]

Complete the following activity to draw tangents to the circle.

1. Draw a circle with radius 3.3 cm and center O. Draw chord PQ of length 6.6 cm. Draw ray OP and ray OQ.
2. Draw a line perpendicular to the ray OP from P.
3. Draw a line perpendicular to the ray OQ from Q.

Some plastic balls of radius 1 cm were melted and cast into a tube. The thickness, length and outer radius of the tube were 2 cm, 90 cm, and 30 cm respectively. How many balls were melted to make the tube?

If sec θ = `41/40`, then find values of sin θ, cot θ, cosec θ

In Quadrilateral ABCD, side AD || BC, diagonal AC and BD intersect in point P, then prove that `"AP"/"PD" = "PC"/"BP"`

Prove the following theorems:

Opposite angles of a cyclic quadrilateral are supplementary.

As shown in figure, LK = `6sqrt(2)` then

1. MK = ?
2. ML = ?
3. MN = ?

Show that points A(– 4, –7), B(–1, 2), C(8, 5) and D(5, – 4) are the vertices of a parallelogram ABCD

Prove that `"cot A"/(1 - tan "A") + "tan A"/(1 - cot"A")` = 1 + tan A + cot A = sec A . cosec A + 1

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

In the given figure, square ABCD is inscribed in the sector A - PCQ. The radius of sector C - BXD is 20 cm. Complete the following activity to find the area of shaded region