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In the figure, \(\angle M=\angle N=46^o \). Express \(x \) in terms of a, b and c, where a, b and c are lengths of \(LM, MN \) and \(NK \) respectively.
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In and
(Corresponding angle)
(Common)
(AA similarly)
In the figure, LM  CB and LN  CD, prove that \(\dfrac{AM}{AB}=\dfrac{AN}{AD} \).
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(1) In
…(1)
In
….(2)
From (1) and (2)
In the given figure, ABC and AMP are two right triangles, right angled at B and M respectively. Prove that :
(i) \(\triangle ABC\sim\triangle AMP \)
(ii) \(\dfrac{CA}{PA}=\dfrac{BC}{MP} \).
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In and
(each )
(common)
(AA similarly)
(Corresponding sides of similar triangles are proportional)
In the figure, DE  OQ and DF  OR. Show that EF  QR.
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In
…(1) (using BPT)
In
…(2) (using BPT)
comparing (1) and (2)
(by converse of BPP)
By using the converse of the basic proportionality theorem, show that the line joining midpoints of nonparallel sides of a trapezium is parallel to the parallel sides.
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In and
( is mid point of )
(alt opp )
(ASA)
and In as and
i.e.
Hence
If three or more parallel lines are intersected by two transversals, the intercepts made by them on the transversal are proportional. Prove.
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In
(using BPT) ….(1)
In
…(2) (using BPT)
From (1) and (2)
(Proved)
\(ABC \) is a triangle in which \(\angle A=90^o,\ AN\bot BC,\ AB=12\ cm \) and \(AC=5\ cm \). Find the ratio of the area of \(\triangle ANC \) and \(\triangle ABC \).
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In and
(common)
(each )
(AA similarly)
ABCD is a square. F is the midpoint of AB.BE is the onethird of BC. If the area of the \(\triangle \)BFE is \(108\ cm^2\), find the length of AC.
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Area of
In the given figure, \(\angle B<90^o \) and segment \(AD\bot BC \), show that \(b^2=h^2+a^2+x^22ax \)
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In an equilateral triangle, prove that three times the square of one side is equal to four times the square of one of its altitudes.
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( is equilateral)
(common)
(AAS)
In
Using BPT, prove that a line drawn through the midpoint of one side of a triangle parallel to another side bisects the third side.
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(using BPT)
or is the mid point of
If \(\triangle ABC\sim\triangle DEF \) and, \(AL \) and \(DM \) are their corresponding medians, then show that \(\dfrac{AL}{DM}=\dfrac{AB}{DE}=\dfrac{BC}{EF}=\dfrac{AC}{DF} \).
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(given)
(AA)
….(1)
In and
(given)
(AA)
….(2)
From (1) and (2)
If \(\triangle ABC\sim\ \triangle DEF \), then show that \(\dfrac{perimeter\ of\ \triangle ABC}{perimeter\ of\ \triangle DEF}=\dfrac{AB}{DE}=\dfrac{BC}{EF}=\dfrac{AC}{DF} \).
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….(1)
…(2)
….(3)
Diagonals of a trapezium ABCD with AB  DC intersect each other at the point O. If AB = 3CD, find the ratio of the areas of triangles AOB and COD.
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In and
(AA)
In \(\triangle \)ABC, in figure, PQ meets AB in P and AC in Q. If If AP = 1 cm, PB = 3 cm, AQ = 1.5 cm, QC = 4.5 cm, prove that area of \(\triangle \)APQ is one sixteenth of the area of \(\triangle \)ABC.
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In and
(common)
(SAS)
In the figure, PA, QB and RC are perpendiculars to AC. Prove that \(\dfrac{1}{x}+\dfrac{1}{y}=\dfrac{1}{z} \)
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Let
In and
(common)
(each )
(AA)
…(1)
Similarly (AA)
…(2)
Adding eqn (1) and (2)
In an equilateral triangle \(ABC, D\) is a point on side \(BC \) such that \(3BD = BC\). Prove that \(9AD^2=7AB^2\).
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In and
(common)
(RHS)
(cpct)
In
In \(\triangle \)PQR,PD \(\bot \) QR such that D lies on QR. If PQ = a, PR = b, QD = C and DR = d and a, b, c, d are positive units, prove that (a+b) (ab) = (c+d) (cd).
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In
….(1)
In
…(2)
From (1) and (2)
In figure, \(BL \) and \(CM \) are medians of \(\triangle ABC \). right angled at \(A \). Prove that \(4(BL^2+CM^2) = 5BC^2\)
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In
In
In the figure, ABC is an isosceles triangle right angled at B. Two equilateral triangles are constructed with side BC and AC. Prove that : \(\mathrm{ar\ \triangle BCD=\dfrac{1}{2}ar\ \triangle ACE} \)
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…(1)
In figure, ABC is right triangle right angled at C. Let BC = a, CA = b, AB = c and let p be the length of perpendicular from C on AB. Prove that
(i) \(cp=ab \)
(ii) \(\dfrac{1}{p^2}=\dfrac{1}{a^2}+\dfrac{1}{b^2} \)
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Area of
Area of
So,
In the figure, PQR is a triangle in which \(QM\bot PR \) and \(PR^2PQ^2=QR^2 \), prove that \(QM^2=PM\times MR \).
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….(1)
In
…(2)
From (1) and (2)
In figure, DE  BC and AD : DB = 5 : 4. Find \(\dfrac{ar(\triangle DFE)}{ar(\triangle CFB)} \).
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…(1)
In and
(alt. )
(VOA)
In the figure, \(PQR \) is a right angled triangle in which \(Q=90^o \). If \(QS=SR \), show that \(PR^24PS^2=3PQ^2 \).
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In the figure, l  m and line segments AB, CD and EF are concurrent at P. Prove that : \(\dfrac{AE}{BF}=\dfrac{AC}{BD}=\dfrac{CE}{FD} \).
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In and
(VOA)
(alt )
(AAA)
…(1)
In and
(VOA)
(alt. )
(AA)
….(2)
Similarly ….(3)
From (1), (2), (3)
In the figure, \(\angle QPR=90^o,\angle PMR=90^o,\ QR=26\ cm,\ PM=8\ cm,\ MR=6\ cm \). Find area \((\triangle PQR) \).
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In
Area of
In \(\triangle \)ABC,D and E are two points lying on side AB such that AD = BE. If DPBC and EQ  AC, then prove that PQAB.
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In ,
and and
and
So by the converse of BPT
Prove that the sum of the squares of the sides of a rhombus is equal to the sum of the squares of its diagonals.
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and
In
In the figure, \(DEFG \) is a square and \(\angle BAC=90^o \). Show that \(DE^2=BD\times EC \).
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In and
(Corresponding angles)
(AA) …(1)
In and
(corresponding angles)
(AA)….(2)
From (1) and (2)
(DEFG is a square)
In the figure, \(\triangle PQR \) is right angled at \(Q \) and the points \(S \) and \(T \) trisect the side \(QR \). Prove that \(8PT^2=3PR^2 + 5PS^2 \).
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Let
and
In
In
In
In the given figure, O is a point in the interior of a triangle ABC, OD \(\bot \) BC,OE \(\bot \) AC and OF \(\bot \) AB. Show that
(i) \(OA^2+OB^2+OC^2OD^2OE^2OF^2= AF^2+ BD^2+CE^2 \)
(ii) \(AF^2 + BD^2 + CE^2 = AE^2 + CD^2 + BF^2 \)
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…(1)
…(2)
…(3)
Adding (1), (2), (3) we get
(Proved)
(ii) From (1) we have
(Proved)
In a right triangle ABC, right angled at C. P and Q are points on the sides CA and CB respectively which divide these sides in the ratio 1 : 2. Prove that :
(i) \( 9AQ^2 = 9AC^2 + 4BC^2 \)
(ii) \(9BP^2 = 9BC^2 + 4AC^2 \)
(iii) \(9(AQ^2 + BP^2) = 13AB^2 \)
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(i)
(Proved) ….(1)
(ii) In
….(2)
(ii) Adding (1) and (2)
In the given figure, in \(\triangle \)PQR XY  QR. PX = 1 cm, XQ = 3 cm, YR = 4.5 cm and QR = 9 cm. Find PY and XY. Further if the area of \(\triangle \)PXY is ‘A’ \(cm^2 \), find in terms of A the area of \(\triangle \)PQR and the area of trapezium XYRQ.
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In and
(corresponding angles)
(AA)
If A be the area of a right triangle and a one of the sides containing the right angle, prove that the length of the altitude on the hypotenuse is \(\dfrac{2Aa}{\sqrt{a^4+4A^2}} \).
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Base of the right angled unit
Area
Altitude
Prove that if a line is drawn parallel to one side of a triangle to intersect the other sides in distinct points, the other two sides are divided in the same ratio.
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Area of
area of ….(1)
Also area of …(2)
Similarly …(3)
….(4)
Dividing (1) by (3) we get
…(5)
Again dividing (2) by (4) we get
….(6)
…(7)
From (5), (6), (7) we get (Proved)
State and prove converse of Pythagoras theorem.
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Area of
area of ….(1)
Also area of …(2)
Similarly …(3)
….(4)
Dividing (1) by (3) we get
…(5)
Again dividing (2) by (4) we get
….(6)
…(7)
From (5), (6), (7) we get (Proved)
By BPT
Adding on both the sides
is isosceles
Prove that if a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.
Using the above result do the following :
In the figure, DE  BC and BD = CE.
Prove that \(\triangle \)ABC is an isosceles triangle.
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In and
(alt )
(AAS)
(cpct)
( is a parallelogram)
In and
(VOA)
(alt )
(AA)
Through the midpoint M of the side CD of a parallelogram ABCD, the line BM is drawn intersecting AC in L and AD produced in E. Prove that EL = 2BL.
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In a triangle if the square of one side is equal to the sum of the square of the other two sides then the angle opposite to the first side is a right angle
To prove
Proof
…(1) ( Since and )
But (given) …(2)
From (1) and (2)
In and