SSC (Marathi Semi-English) इयत्ता १० वी - Maharashtra State Board Important Questions

Given: In the figure, point A is in the exterior of the circle with centre P. AB is the tangent segment and secant through A intersects the circle in C and D.

To prove: AB² = AC × AD

Construction: Draw segments BC and BD.

Write the proof by completing the activity.

Proof: In ΔABC and ΔADB,

∠BAC ≅ ∠DAB  .....becuase ______

∠______ ≅ ∠______  ......[Theorem of tangent secant]

∴ ΔABC ∼ ΔADB  .......By ______ test

∴ `square/square = square/square`   .....[C.S.S.T.]

∴  AB² = AC × AD

Proved.

In the figure, the centre of the circle is O and ∠STP = 40°.

1. m (arc SP) = ? By which theorem?
2. m ∠SOP = ? Give reason.

Find the value of y, if the points A(3, 4), B(6, y) and C(7, 8) are collinear.

In the following figure, a quadrilateral LMNO circumscribes a circle with centre C. ∠O = 90°, LM = 25 cm, LO = 27 cm and MJ = 6 cm. Calculate the radius of the circle.

A pizza has 8 slices all equally spaced. Suppose pizza is a flat circle of radius 28 cm, find the area covered between 3 slices of pizza.

In the above figure, ∠L = 35°, find :

1. m(arc MN)
2. m(arc MLN)

Solution :

1. ∠L = `1/2` m(arc MN) ............(By inscribed angle theorem)
   ∴ `square = 1/2` m(arc MN)
   ∴ 2 × 35 = m(arc MN)
   ∴ m(arc MN) = `square`
2. m(arc MLN) = `square` – m(arc MN) ...........[Definition of measure of arc]
   = 360° – 70°
   ∴ m(arc MLN) = `square`

In the above figure, ∠ABC is inscribed in arc ABC.

If ∠ABC = 60°. find m ∠AOC.

Solution:

∠ABC = `1/2` m(arc AXC)   ......`square`

60° = `1/2` m(arc AXC) 

`square` = m(arc AXC) 

But m ∠AOC = \[\boxed{m(arc ....)}\]   ......(Property of central angle)

∴ m ∠AOC = `square`

Construct the circumcircle and incircle of an equilateral triangle ABC with side 6 cm and centre O. Find the ratio of radii of circumcircle and incircle.

Draw `angle ABC` of measure 80° and bisect it

Draw ∠ABC of measures 135^°and bisect it.

∆AMT ~ ∆AHE. In ∆AMT, AM = 6.3 cm, ∠TAM = 50°, AT = 5.6 cm. `"AM"/"AH" = 7/5`. Construct ∆AHE.

∆ABC ~ ∆LBN. In ∆ABC, AB = 5.1 cm, ∠B = 40°, BC = 4.8 cm, \[\frac{AC}{LN} = \frac{4}{7}\]. Construct ∆ABC and ∆LBN.

Construct ∆PYQ such that, PY = 6.3 cm, YQ = 7.2 cm, PQ = 5.8 cm. If \[\frac{YZ}{YQ} = \frac{6}{5},\] then construct ∆XYZ similar to ∆PYQ.

If A(–14, –10), B(6, –2) is given, find the coordinates of the points which divide segment AB into four equal parts.

If A (20, 10), B(0, 20) are given, find the coordinates of the points which divide segment AB into five congruent parts.

The line segment AB is divided into five congruent parts at P, Q, R and S such that A–P–Q–R–S–B. If point Q(12, 14) and S(4, 18) are given find the coordinates of A, P, R, B.

Ponit M is the mid point of seg AB and AB = 14 then AM = ?

Observe the adjoining figure and write down one pair of interior angles.