Diagonals AC and BD of a quadrilateral ABCD intersect at O in such a way that ar(△AOD) = ar(△BOC) . Prove that ABCD is a trapezium. Solution Given: ar(ΔAOD) = ar(ΔBOC) Prove: ABCD is a trapezium. Proof: ar(ΔAOD) = ar(ΔBOC) => ar(ΔAOD)... View Article
In Fig. ABC and ABD are two triangles on the same base AB. If line- segment CD is bisected by AB at O, show that: ar(ABC) = ar(ABD) . Solution In ΔABC, AO is the median. (CD is bisected by AB at O) Hence, ar(AOC) = ar(AOD) ---------------(1) also, ΔBCD, BO is... View Article
A stone is released from the top of a tower of height 19.6 m. Calculate its final velocity just before touching the ground. Given that Initial velocity u = 0 Tower height = total distance = 19.6m g = 9.8 m/s2 Find out Final velocity just before touching the... View Article
Draw different pairs of circles. How many points does each pair have in common? What is the maximum number of common points? In these two circles, there is no point in common. Here, only one point “P” is common. Even here, P is the common point.... View Article
A die is thrown once. Find the probability of getting (i) a prime number; (ii) a number lying between 2 and 6; (iii) an odd number. Given that A die is thrown once Find out The probability of getting (i) a prime number; (ii) a number lying between 2 and 6; (iii) an odd number.... View Article
ABCD is a trapezium with AB || DC. A line parallel to AC intersects AB at X and BC at Y. Prove that ar (△ADX) = ar (△ACY) . [Hint : Join CX.] Solution ABCD is a trapezium with,AB || DC, XY || AC Construction Join CX To Prove: ar(ADX) = ar(ACY) Proof: ar(ΔADX) =... View Article
Sides of two similar triangles are in the ratio 4 : 9. Areas of these triangles are in the ratio (A) 2 : 3 (B) 4 : 9 (C) 81 : 16 (D) 16 : 81 The correct answer is (D). The sides of two similar triangles are in the ratio 4: 9. Let us assume that ABC and DEF are two similar... View Article
A ladder 10 m long reaches a window 8 m above the ground. Find the distance of the foot of the ladder from base of the wall. Given A ladder 10 m long reaches a window 8 m above the ground. Find out The distance of the foot of the ladder from base of the wall.... View Article
In the figure, ABC and DBC are two triangles on the same base BC. If AD intersects BC at O, show that area (ΔABC) /area (ΔDBC) = AO/DO. Given that ABC and DBC are two triangles on the same base BC. AD intersects BC at O. To Prove Area (ΔABC)/Area (ΔDBC) = AO/DO... View Article
Sum of the areas of two squares is 468 m2. If the difference of their perimeters is 24 m, find the sides of the two squares. Let the sides of the two squares be x m and y m. Therefore, their perimeter will be 4x and 4y respectively And area... View Article
The length of a tangent from a point A at distance 5 cm from the centre of the circle is 4 cm. Find the radius of the circle. OB⊥ AB then ∆OAB is a right-angled triangle OA2 = AB2 + OB2 (Pythagoras Theorem) 52 = 42 + OB2 OB2 =... View Article
A right triangle ABC with sides 5cm, 12cm and 13cm is revolved about the side 12 cm. Find the volume of the solid so obtained. Given Radius of cone (r) =5 cm Height of cone (h) = 12 cm Volume of cone = (1/3) × πr2h = (1/3) × π × 5 × 5... View Article
ΔABC is an isosceles triangle in which AB = AC. Side BA is produced to D such that AD = AB (see Fig.) . Show that BCD is a right angle. Solution Given AB = AC and AD = AB To Prove BCD is a right angle Proof In ∆ABC, AB=AC ∠ACB=∠ABC --------------------(i) In... View Article
XY is a line parallel to side BC of a triangle ABC. If BE || AC and CF || AB meet XY at E and F respectively, show that ar(ΔABE) = ar(ΔACF) Solution Given XY || BC, BE || AC and CF || AB To Prove ar(ΔABE) = ar(ΔAC) Proof: BCYE is a parallelogram as ΔABE and... View Article
In Fig., if TP and TQ are the two tangents to a circle with centre O so that ∠POQ = 110°, then ∠PTQ is equal to (A) 60° (B) 70° (C) 80° (D) 90° Option (B) 70° In the quadrilateral OPTQ, ∠P = 90°, ∠Q = 90° (Theorem 10.1) ∠O = 110° The sum of the... View Article
In Fig., ABCDE is a pentagon. A line through B parallel to AC meets DC produced at F. Show that (i) ar(△ACB) = ar(△ACF) (ii) ar(AEDF) = ar(ABCDE) Solution (i) ΔACB and ΔACF lie on the same base AC and between the same parallels AC and BF. Hence, ar(ΔACB) =... View Article
Prove that the circle drawn with any side of a rhombus as diameter, passes through the point of intersection of its diagonals. Solution To prove: A circle drawn with Q as centre, will pass through A, B and O (i.e. QA = QB = QO) Proof We know that all sides... View Article
If each edge of a cube is doubled, (i) how many times will its surface area increase? (ii) how many times will its volume increase? Let us assume that the original edge of the cube is x cm. If the edge of the cube is doubled, then the new edge will... View Article
ABCD is a cyclic quadrilateral whose diagonals intersect at a point E. If ∠ DBC = 70°, ∠ BAC is 30°, find ∠ BCD. Further, if AB = BC, find ∠ ECD. Solution Angles in the same segment are equal. So, ∠ CBD = ∠ CAD ∴ ∠ CAD = 70° ∠ BAD will be equal to the sum of... View Article
Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle. As the tangents from the point are equal AP = AS OA = OA (It is the common side) OP = OS (They are the radii of the circle) So, by SSS... View Article
