mathemerize

Solution : Given : ABC is a triangle and D is point on BC such that \(BD\over CD\) = \(AB\over AC\) To Prove : AD is the bisector of \(\angle\) BAC. Construction : Produce line BA to E such that line AE = AC. Join CE. Proof : In \(\triangle\) AEC, since AE = AC, …

In the figure, D is a point on side BC of triangle ABC such that \(BD\over CD\) = \(AB\over AC\). Prove that AD is the bisector of \(\angle\) BAC. Read More »

Solution : (i)  In \(\triangle\)s PAC and PDB, we have : \(\angle\) APC = \(\angle\) DPB         (common) \(\angle\) PAC = \(\angle\) PDB [\(\therefore\)  \(\angle\) BAC = 180 – \(\angle\) PAC  and \(\angle\) PDB = \(\angle\) CDB = 180 – (180 – \(\angle\) PAC) = \(\angle\) PAC] \(\therefore\)  By AA similarity, we have …

In the figure, two chords AB and CD of a circle intersect each other at the point P (when produced) outside the circle. Prove that (i) \(\triangle\) PAC ~ \(\triangle\) PDB (ii) PA.PB = PC.PD Read More »

Solution : (i)  In \(\triangle\)s PAC and PDB, we have : \(\angle\) APC = \(\angle\) DPB         (vertically opp. angles) \(\angle\) CAP = \(\angle\) BDP        (angles in same segment of circle are equal) \(\therefore\)  By AA similarity, we have : \(\triangle\) APC ~ \(\triangle\) DPB  (ii)  Since \(\triangle\)s APC ~ …

In the figure, two chords AB and CD intersect each other at the point P. Prove that (i) \(\triangle\) APC ~ \(\triangle\) DPB (ii) AP.PB = CP.DP Read More »

Solution : We know that AD is a median of triangle ABC, then \({AB}^2\) + \({AC}^2\) = 2\({AD}^2\) + \({1\over 2}{BC}^2\) Since the diagonals of a parallelogram bisect each other, therefore BO and DO are medians of triangles ABC and ADC respectively. \(\therefore\)  \({AB}^2\) + \({BC}^2\) = 2\({BO}^2\) + \({1\over 2}{AC}^2\)          …

Prove that the sum of the squares of the diagonals of a parallelogram is equal to the sum of the squares of its sides. Read More »

Solution : Given : ABC is a triangle and AD is the median. To Prove : (i)  \({AB}^2\) + \({AC}^2\) = 2[\({AD}^2\) + \({BD}^2\)] (ii)  \({AB}^2\) + \({AC}^2\) = 2\({AD}^2\) + 2\([{1\over 2}BC]^2\) Construction : Draw AE \(\perp\) BC Proof : In right angled triangle ABE, by Pythagoras theorem, \({AB}^2\) = \({BE}^2\) + \({AE}^2\)    …

Prove that in any triangle, the sum of the squares of any two sides is equal to twice the square of half of the third side together with twice the square of the median which bisects the third side. Read More »

Solution : Given : In triangle ABC angle B is an acute angle and AD \(\perp\) BC To Prove : \({AC}^2\) = \({AB}^2\) + \({BC}^2\) – 2BC.BD Proof : Since \(\triangle\) ADC is a right triangle, angled at D. Therefore, by Pythagoras theorem, \({AC}^2\) = \({AD}^2\) + \({CD}^2\) \(\implies\)  \({AC}^2\) = \({AD}^2\) + \({BC – …

In an acute triangle, the square of the side opposite to an acute angle is equal to the sum of the squares of other two sides minus twice the product of one side and the projection of the other on first. Read More »

Solution : Given : ABC is triangle in which \(\angle\) ABC > 90 and AD \(\perp\) CB produced. To Prove : \({AC}^2\) = \({AB}^2\) + \({BC}^2\) + 2BC.BD Proof : Since \(\triangle\) ADB is a right triangle, angled at D. Therefore, by Pythagoras theorem, \({AB}^2\) = \({AD}^2\) + \({DB}^2\)           ……..(1) Again, …

In the figure, ABC is a triangle in which \(\angle\) ABC > 90 and AD \(\perp\) CB produced. Prove that \({AC}^2\) = \({AB}^2\) + \({BC}^2\) + 2BC.BD. Read More »

Solution : We have : AB \(\perp\) BC and DM \(\perp\) BC. So,        AB || DM Similarly, we have : BC \(\perp\) AB and DN \(\perp\) AB. So,          CB || DN Hence, quadrilateral BMDN is a rectangle. \(\therefore\)  BM = DN (i)  In triangle BMD, we have : …

In the figure, D is a point on hypotenuse AC of \(\triangle\) ABC, BD \(\perp\) AC, DM \(\perp\) BC and DN \(\perp\) AB. Prove that (i) \({DM}^2\) = DN.MC (ii) \({DN}^2\) = DM.AN Read More »

Solution : Given : PQR is a triangle and PS is the bisector of \(\angle\) QPR meeting QR at S. \(\therefore\)  \(\angle\) QPS = \(\angle\) SPR To Prove : \(QS\over SR\) = \(PQ\over PR\) Construction : Draw RT parallel to SP to cut QP produced at T. Proof : Since PS || TR and PR …

In the figure, PS is the bisector of \(\triangle\) PQR. Prove that \(QS\over SR\) = \(PQ\over PR\). Read More »

Solution : In triangle ABC, we have : AB = \(6\sqrt{3}\) cm , AC = 12 cm and BC = 6 cm Now, \({AB}^2\) + \({BC}^2\) = \((6\sqrt{3})^2\) + \((6)^2\) = \(36 \times 3\) + 36 = 108 + 36 = 144 = \((AC)^2\) Thus, triangle ABC is a right angled triangle at B. \(\therefore\)  …

Tick the correct answer and justify : In \(\triangle\), AB = \(6\sqrt{3}\) cm, AC = 12 cm and BC = 6 cm. The angle is : (i) 120 (ii) 60 (iii) 90 (iv) 45 Read More »