In a quadrilateral ABCD, if AO and BO are the bisectors of ∠A and ∠B respectively, ∠C = 70°
Question: In a quadrilateralABCD, ifAOandBOare the bisectors ofAandBrespectively, C= 70 and D= 30. Then, AOB= ?(a) 40(b) 50(c) 80(d) 100 Solution: (b) 50oExplanation: C= 70oandD=30oThenA+ B=360o- (70 +30)o= 260o $\therefore \frac{1}{2}(\angle A+\angle B)=\frac{1}{2}\left(260^{\circ}\right)=130^{\circ}$ In∆AOB, we have: $\angle A O B=180^{\circ}-\left[\frac{1}{2}(\angle A+\angle B)\right]$ ⇒AOB =180 - 130 =50o...
Read More →If ABCD is a parallelogram with two adjacent angles ∠A = ∠B,
Question: IfABCDis a parallelogram with two adjacent anglesA= B, then the parallelogram is a(a) rhombus(b) trapezium(c) rectangle(d) none of these Solution: (c) RectangleExplanation:A= BThenA+ B=180o⇒2A=180o⇒ A =90o⇒ A = B =C =D =90o The parallelogram is a rectangle....
Read More →On a horizontal plane there is vertical tower with a flag pole on the top of the tower.
Question: On a horizontal plane there is vertical tower with a flag pole on the top of the tower. At a point 9 metres away from the foot of the tower the angle of elevation of the top and bottom of the flag pole are 60 and 30 respectively. Find the height of the tower and the flag pole mounted on it. Solution: LetABbe the tower of heighthandADbe the flag pole on tower. At the point 9m away from the foot of tower, the angle of elevation of the top and bottom of flag pole are 60 and 30. LetAD = x,...
Read More →A diagonal of a rectangle is inclined to one side of the rectangle at 35°.
Question: A diagonal of a rectangle is inclined to one side of the rectangle at 35. The acute angle between the diagonals is(a) 55(b) 70(c) 45(d) 50 Solution: Given: In rectangleABCD,OAD=35.Since,BAD =90⇒⇒OAB=90-35=55InΔOAB,Since,OA=OB (Diagonals of a rectangle are equal and bisect each other)⇒⇒OAB=OBA=55 (Angles opposite to equal sides are equal)Now, inΔODA,55+55+DOA=180 (Anglesumproperty of a triangle)⇒⇒DOA=180-110⇒⇒DOA=70Thus, theacuteanglebetweenthediagonalsis70.Hence, the correct option is ...
Read More →The length of each side of a rhombus is 10 cm and one if its diagonals is of length 16 cm.
Question: The length of each side of a rhombus is 10 cm and one if its diagonals is of length 16 cm. The length of the other diagonal is(a) 13 cm(b) 12 cm (c) $2 \sqrt{39} \mathrm{~cm}$ (d) $6 \mathrm{~cm}$ Solution: (b) 12 cmExplanation: Let $A B C D$ be the rhombus. $\therefore A B=B C=C D=D A=10 \mathrm{~cm}$ LetACandBDbe the diagonals of the rhombus. LetACbexandBDbe16 cm and O be the intersection point of the diagonals. We know that the diagonals of a rhombus are perpendicular bisectors of e...
Read More →Evaluate each of the following:
Question: Evaluate each of the following: (i) $\cot ^{-1}\left(\cot \frac{\pi}{3}\right)$ (ii) $\cot ^{-1}\left(\cot \frac{4 \pi}{3}\right)$ (iii) $\cot ^{-1}\left(\cot \frac{9 \pi}{4}\right)$ (iv) $\cot ^{-1}\left(\cot \frac{19 \pi}{6}\right)$ (v) $\cot ^{-1}\left\{\cot \left(-\frac{8 \pi}{3}\right)\right\}$ (vi) $\cot ^{-1}\left\{\cot \left(\frac{21 \pi}{4}\right)\right\}$ Solution: We know that $\cot ^{-1}(\cot \theta)=\theta, \quad(0, \pi)$ (i) We have $\cot ^{-1}\left(\cot \frac{\pi}{3}\rig...
Read More →The lengths of the diagonals of a rhombus are 16 cm and 12 cm.
Question: The lengths of the diagonals of a rhombus are 16 cm and 12 cm. The length of each side of the rhombus is(a) 10 cm(b) 12 cm(c) 9 cm(d) 8 cm Solution: (a) 10 cmExplanation: LetABCDbe the rhombus.AB = BC = CD = DAHere, ACandBDare the diagonals ofABCD,whereAC= 16 cm andBD= 12 cm.Let the diagonals intersect each other at O.We know that the diagonals of a rhombus are perpendicular bisectors of each other.∆AOBis a right angle triangle, in whichOA = AC/2 = 16/2 = 8 cm andOB = BD/2 = 12/2 = 6...
Read More →If the diagonals of a quadrilateral bisect each other at right angles,
Question: If the diagonals of a quadrilateral bisect each other at right angles, then the figure is a(a) trapezium(b) parallelogram(c) rectangle(d) rhombus Solution: (d) rhombusThe diagonals of a rhombus bisect each other at right angles....
Read More →In which of the following figures are the diagonals equal?
Question: In which of the following figures are the diagonals equal?(a) Parallelogram(b) Rhombus(c) Trapezium(d) Rectangle Solution: (d) Rectangle.The diagonals of a rectangle are equal....
Read More →ABCD is a rhombus such that ∠ACB = 50°. Then, ∠ADB = ?
Question: ABCDis a rhombus such that ACB= 50. Then, ADB= ?(a) 40(b) 25(c) 65(d) 130 Solution: Weknowthatdiagonalsofrhombusbisecteachotherat90.Then, inΔBOC,90+50+OBC=180 (Angle sum property of triangle)⇒⇒OBC=180-140⇒⇒OBC=40ButOBC =ADB (Alternate interior angles)Thus,ADB=40Hence, the corerct option is (a)....
Read More →In the given figure, ABCD is a parallelogram in which ∠BAD = 75° and ∠CBD = 60°.
Question: In the given figure,ABCDis a parallelogram in whichBAD= 75 and CBD= 60. Then, BDC= ?(a) 60(b) 75(c) 45(d) 50 Solution: (c)45Explanation:B = 180o A⇒ B = 180o 75o= 105oNow, B =ABD +CBD⇒ 105o =ABD + 60o⇒ ABD = 105o 60o= 45o⇒ ABD =BDC = 45o (Alternate angles)...
Read More →The angles of a quadrilateral are in the ratio 3 : 4 : 5 : 6.
Question: The angles of a quadrilateral are in the ratio 3 : 4 : 5 : 6. The smallest of these angles is(a) 45(b) 60(c) 36(d) 48 Solution: (b)60Explanation:Let A =3x, B =4x, C =5xandD =6x.Since the sum of the angles of a quadrilateral is360o, we have:3x+4x+5x+6x=360o⇒18x=360o⇒x= 20o A= 60o, B= 80o, C=100oandD=120oHence, the smallest angle is60....
Read More →Three angles of a quadrilateral are 80°, 95° and 112°. Its fourth angle is
Question: Three angles of a quadrilateral are 80, 95 and 112. Its fourth angle is(a) 78(b) 73(c) 85(d) 100 Solution: (b)73Explanation:Let the measure of the fourth angle bexo.Since the sum of the angles of a quadrilateral is360o, we have:80o+ 95o+ 112o+x= 360o⇒287o+x= 360o⇒x= 73oHence, the measure of the fourth angle is 73o....
Read More →From the top of a building 15 m high the angle of elevation of the top of a tower is found to be 30°.
Question: From the top of a building 15 m high the angle of elevation of the top of a tower is found to be 30. From the bottom of the same building, the angle of elevation of the top of the tower is found to be 60. Find the height of the tower and the distance between the tower and building. Solution: In the figure letOD=handADbe the tower. The angle of elevation from the top of building to the top of tower is to be found 30. Height of building ism and an angle of elevation from the bottom of sa...
Read More →The midpoints of the sides AB, BC, CD and DA of a quadrilateral ABCD are joined to form a quadrilateral.
Question: The midpoints of the sidesAB,BC,CDandDAof a quadrilateralABCDare joined to form a quadrilateral. IfAC=BDandACBDthen prove that the quadrilateral formed is a square. Solution: Given: In quadrilateralABCD,AC=BDandACBD.P, Q, RandSarethemid-pointsofAB, BC,CDandAD,respectively.Toprove:PQRSis a square.Construction: JoinACandBD.Proof:InΔABC,∵∵PandQaremid-pointsofABandBC, respectively. $\therefore P Q \| A C$ and $P Q=\frac{1}{2} A C$ (Mid-point theorem) ...(1) Similarly, inΔACD,∵∵RandSaremid-...
Read More →The angle of elevation of the top of a tower from a point A on the ground is 30°.
Question: The angle of elevation of the top of a tower from a point A on the ground is 30. On moving a distance of 20 metres towards the foot of the tower to a pointBthe angle of elevation increases to 60. Find the height of the tower and the distance of the tower from the pointA. Solution: Letbe height of tower and the angle of elevation of the top of tower from a pointon the ground isand on moving with distancem towards the foot of tower on the pointis. Letand Now we have to find height of tow...
Read More →The diagonals of a quadrilateral ABCD are perpendicular to each other.
Question: The diagonals of a quadrilateralABCDare perpendicular to each other. Prove that the quadrilateral formed by joining the midpoints of its sides is a rectangle. Solution: Given: In quadrilateralABCD,ACBD.P, Q, RandSarethemid-pointsofAB, BC,CDandAD,respectively.Toprove:PQRSis a rectangle.Proof:InΔABC,PandQaremid-pointsofABandBC,respectively. $\therefore P Q \| A C$ and $P Q=\frac{1}{2} A C$ (Mid-point theorem) ...(1) Similarly, inΔACD,So,RandSaremid-pointsofsidesCDandAD, respectively. $\t...
Read More →Evaluate each of the following:
Question: Evaluate each of the following: (i) $\operatorname{cosec}^{-1}\left(\operatorname{cosec} \frac{\pi}{4}\right)$ (ii) $\operatorname{cosec}^{-1}\left(\operatorname{cosec} \frac{3 \pi}{4}\right)$ (iii) $\operatorname{cosec}^{-1}\left(\operatorname{cosec} \frac{6 \pi}{5}\right)$ (iv) $\operatorname{cosec}^{-1}\left(\operatorname{cosec} \frac{11 \pi}{6}\right)$ (v) $\operatorname{cosec}^{-1}\left(\operatorname{cosec} \frac{13 \pi}{6}\right)$ (vi) $\operatorname{cosec}^{-1}\left\{\operatorna...
Read More →The angle of elevation of the top of a tower as observed form a point in a horizontal plane through
Question: The angle of elevation of the top of a tower as observed form a point in a horizontal plane through the foot of the tower is 32. When the observer moves towards the tower a distance of 100 m, he finds the angle of elevation of the top to be 63. Find the height of the tower and the distance of the first position from the tower. [Take tan 32 = 0.6248 and tan 63 = 1.9626] Solution: Lethbe height of tower and the angle of elevation as observed from the foot of tower is 32 and observed move...
Read More →The diagonals of a quadrilateral ABCD are equal.
Question: The diagonals of a quadrilateralABCDare equal. Prove that the quadrilateral formed by joining the midpoints of its sides is a rhombus. Solution: Given: InquadrilateralABCD,BD=ACandK,L,MandNare the mid-points ofAD,CD,BCandAB, respectively.Toprove:KLMNisarhombus.Proof:In∆ADC,Since,KandLare themid-points ofsidesADandCD, respectively. So, $K L \| A C$ and $K L=\frac{1}{2} A C$ ...(1) Similarly, in∆ABC,Since,MandNare themid-pointsofsidesBCandAB, respectively. So, $N M \| A C$ and $N M=\fra...
Read More →Evaluate each of the following:
Question: Evaluate each of the following: (i) $\sec ^{-1}\left(\sec \frac{\pi}{3}\right)$ (ii) $\sec ^{-1}\left(\sec \frac{2 \pi}{3}\right)$ (iii) $\sec ^{-1}\left(\sec \frac{5 \pi}{4}\right)$ (iv) $\sec ^{-1}\left(\sec \frac{7 \pi}{3}\right)$ (v) $\sec ^{-1}\left(\sec \frac{9 \pi}{5}\right)$ (vi) $\sec ^{-1}\left\{\sec \left(-\frac{7 \pi}{3}\right)\right\}$ (vii) $\sec ^{-1}\left(\sec \frac{13 \pi}{4}\right)$ (viii) $\sec ^{-1}\left(\sec \frac{25 \pi}{\pi}\right)$ Solution: We know that $\sec ^...
Read More →The angle of elevation of a tower from a point on the same level as the foot of the tower is 30°
Question: The angle of elevation of a tower from a point on the same level as the foot of the tower is $30^{\circ}$. On advancing 150 metres towards the foot of the tower, the angle of elevation of the tower becomes $60^{\circ}$. Show that the height of the tower is $129.9$ metres (Use $\sqrt{3}=1.732$ ) Solution: Lethbe height of tower and angle of elevation of foot of tower is 30, on advancing150 m towards the foot of tower then angle of elevation becomes 60. We assume thatBC=xandCD= 150 m. No...
Read More →On the same side of a tower, two objects are located.
Question: On the same side of a tower, two objects are located. When observed from the top of the tower, their angles of depression are 45 and 60. If the height of the tower is 150 m, find the distance between the objects. Solution: Let AB be the tower of heightm and Two objects are located when top of tower are observed, makes an angle of depression from the top and bottom of tower areandrespectively. So we use trigonometric ratios. In a triangle ABC, $\tan 45^{\prime}=\frac{150}{x+y}$ $\Righta...
Read More →Prove that the line segments joining the midpoints of opposite sides of a quadrilateral bisect each other.
Question: Prove that the line segments joining the midpoints of opposite sides of a quadrilateral bisect each other. Solution: LetABCDbe the quadrilateral in whichP, Q, R, andSare the midpoints of sidesAB, BC, CD, andDA,respectively.JoinPQ, QR, RS,SPandBD.BDis a diagonal ofABCD. In ΔABD,SandPare the midpoints ofADandAB,respectively. $\therefore S P \| B D$ and $S P=\frac{1}{2} B D$.... (i) (By midpoint theorem) Similarly in ΔBCD,we have: $Q R \| B D$ and $Q R=\frac{1}{2} B D$ ... (ii) (By midpoi...
Read More →Evaluate each of the following:
Question: Evaluate each of the following: (i) $\tan ^{-1}\left(\tan \frac{\pi}{3}\right)$ (ii) $\tan ^{-1}\left(\tan \frac{6 \pi}{7}\right)$ (iii) $\tan ^{-1}\left(\tan \frac{7 \pi}{6}\right)$ (iv) $\tan ^{-1}\left(\tan \frac{9 \pi}{4}\right)$ (v) $\tan ^{-1}(\tan 1)$ (v) $\tan ^{-1}(\tan 2)$ (v) $\tan ^{-1}(\tan 4)$ (v) $\tan ^{-1}(\tan 12)$ Solution: We know that $\tan ^{1}(\tan \theta)=\theta, \quad-\frac{\pi}{2}\theta\frac{\pi}{2}$ (i) We have $\tan ^{-1}\left(\tan \frac{\pi}{3}\right)=\frac...
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