Question and Answers Forum

GeometryQuestion and Answers: Page 110

Question Number 16665 Answers: 0 Comments: 1

Question Number 16641 Answers: 0 Comments: 0

Prove that p is a prime number if and only if every equiangular polygon with p sides of rational lengths is regular.

$$\mathrm{Prove}\:\mathrm{that}\:{p}\:\mathrm{is}\:\mathrm{a}\:\mathrm{prime}\:\mathrm{number}\:\mathrm{if}\:\mathrm{and} \\ $$$$\mathrm{only}\:\mathrm{if}\:\mathrm{every}\:\mathrm{equiangular}\:\mathrm{polygon}\:\mathrm{with} \\ $$$${p}\:\mathrm{sides}\:\mathrm{of}\:\mathrm{rational}\:\mathrm{lengths}\:\mathrm{is}\:\mathrm{regular}. \\ $$

Question Number 16595 Answers: 1 Comments: 4

Question Number 16592 Answers: 1 Comments: 1

please what does the question mean by the overlapping portion of A and B.

$$\mathrm{please}\:\mathrm{what}\:\mathrm{does}\:\mathrm{the}\:\mathrm{question}\:\mathrm{mean} \\ $$$$\mathrm{by}\:\mathrm{the}\:\mathrm{overlapping}\:\mathrm{portion}\:\mathrm{of}\:\mathrm{A}\: \\ $$$$\mathrm{and}\:\mathrm{B}. \\ $$

Question Number 16579 Answers: 0 Comments: 3

Question Number 16570 Answers: 0 Comments: 1

A,B and E are three circles all with radius 1 unit.A and E touch at P whole B and E touch at Q. ∠POQ=x° where O is the centre of E.Find the area of the overlapping portion of A and B if 0≤x≤60°

$$\mathrm{A},\mathrm{B}\:\mathrm{and}\:\mathrm{E}\:\mathrm{are}\:\mathrm{three}\:\mathrm{circles}\:\mathrm{all}\: \\ $$$$\mathrm{with}\:\mathrm{radius}\:\mathrm{1}\:\mathrm{unit}.\mathrm{A}\:\mathrm{and}\:\mathrm{E}\:\mathrm{touch} \\ $$$$\mathrm{at}\:\mathrm{P}\:\mathrm{whole}\:\mathrm{B}\:\mathrm{and}\:\mathrm{E}\:\mathrm{touch}\:\mathrm{at}\:\mathrm{Q}. \\ $$$$\angle\mathrm{POQ}=\mathrm{x}°\:\mathrm{where}\:\mathrm{O}\:\mathrm{is}\:\mathrm{the}\:\mathrm{centre}\: \\ $$$$\mathrm{of}\:\mathrm{E}.\mathrm{Find}\:\mathrm{the}\:\mathrm{area}\:\mathrm{of}\:\mathrm{the}\: \\ $$$$\mathrm{overlapping}\:\mathrm{portion}\:\mathrm{of}\:\mathrm{A}\:\mathrm{and}\:\mathrm{B}\:\mathrm{if} \\ $$$$\mathrm{0}\leqslant\mathrm{x}\leqslant\mathrm{60}° \\ $$

Question Number 16483 Answers: 0 Comments: 12

Question Number 16464 Answers: 0 Comments: 2

Question Number 16409 Answers: 3 Comments: 9

Question Number 16364 Answers: 1 Comments: 0

Question Number 16441 Answers: 1 Comments: 2

Question Number 16302 Answers: 1 Comments: 5

Related to Q16140 What if the three lines d_1 ,d_2 ,d_3 are not parallel, but concurrent?

Question Number 16277 Answers: 3 Comments: 1

Question Number 16226 Answers: 0 Comments: 1

Question Number 16214 Answers: 2 Comments: 4

In ΔABC, r_1 , r_2 and r_3 are the exradii as shown. Prove that r_1 = (Δ/(s − a)) , r_2 = (Δ/(s − b)) and r_3 = (Δ/(s − c)) . Here s = ((a + b + c)/2) .

$$\mathrm{In}\:\Delta{ABC},\:{r}_{\mathrm{1}} ,\:{r}_{\mathrm{2}} \:\mathrm{and}\:{r}_{\mathrm{3}} \:\mathrm{are}\:\mathrm{the}\:\mathrm{exradii} \\ $$$$\mathrm{as}\:\mathrm{shown}.\:\mathrm{Prove}\:\mathrm{that}\:{r}_{\mathrm{1}} \:=\:\frac{\Delta}{{s}\:−\:{a}}\:, \\ $$$${r}_{\mathrm{2}} \:=\:\frac{\Delta}{{s}\:−\:{b}}\:\mathrm{and}\:{r}_{\mathrm{3}} \:=\:\frac{\Delta}{{s}\:−\:{c}}\:.\:\mathrm{Here} \\ $$$${s}\:=\:\frac{{a}\:+\:{b}\:+\:{c}}{\mathrm{2}}\:. \\ $$

Question Number 16194 Answers: 0 Comments: 21

Question Number 16140 Answers: 2 Comments: 0

Question Number 16110 Answers: 0 Comments: 1

let a_1 >a_2 >0 and a_(n+1) =(√(a_n a_(n−1 ) )) where n is greater than equal to 2 Then The sequence {a_(2n) } is (1) monotonic increasing (2)monotonic decreasing (3)non monotonic (4)unbounded

$$\mathrm{let}\:\mathrm{a}_{\mathrm{1}} >\mathrm{a}_{\mathrm{2}} >\mathrm{0}\:\mathrm{and}\:\mathrm{a}_{\mathrm{n}+\mathrm{1}} =\sqrt{\mathrm{a}_{\mathrm{n}} \mathrm{a}_{\mathrm{n}−\mathrm{1}\:\:\:} } \\ $$$$\mathrm{where}\:\mathrm{n}\:\mathrm{is}\:\mathrm{greater}\:\mathrm{than}\:\mathrm{equal}\:\mathrm{to}\:\mathrm{2}\: \\ $$$$\mathrm{Then} \\ $$$$\mathrm{The}\:\mathrm{sequence}\:\left\{\mathrm{a}_{\mathrm{2n}} \right\}\:\mathrm{is}\: \\ $$$$\left(\mathrm{1}\right)\:\mathrm{monotonic}\:\mathrm{increasing} \\ $$$$\left(\mathrm{2}\right)\mathrm{monotonic}\:\mathrm{decreasing} \\ $$$$\left(\mathrm{3}\right)\mathrm{non}\:\mathrm{monotonic} \\ $$$$\left(\mathrm{4}\right)\mathrm{unbounded} \\ $$$$ \\ $$

Question Number 16108 Answers: 1 Comments: 1

Question Number 16077 Answers: 0 Comments: 0

Let ABCDE be an equiangular pentagon whose side lengths are rational numbers. Prove that the pentagon is regular.

$$\mathrm{Let}\:{ABCDE}\:\mathrm{be}\:\mathrm{an}\:\mathrm{equiangular} \\ $$$$\mathrm{pentagon}\:\mathrm{whose}\:\mathrm{side}\:\mathrm{lengths}\:\mathrm{are} \\ $$$$\mathrm{rational}\:\mathrm{numbers}.\:\mathrm{Prove}\:\mathrm{that}\:\mathrm{the} \\ $$$$\mathrm{pentagon}\:\mathrm{is}\:\mathrm{regular}. \\ $$

Question Number 16075 Answers: 0 Comments: 0

Prove that the perpendiculars dropped from the midpoints of the sides of a cyclic quadrilateral to the opposite sides are concurrent.

$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{perpendiculars} \\ $$$$\mathrm{dropped}\:\mathrm{from}\:\mathrm{the}\:\mathrm{midpoints}\:\mathrm{of}\:\mathrm{the}\:\mathrm{sides} \\ $$$$\mathrm{of}\:\mathrm{a}\:\mathrm{cyclic}\:\mathrm{quadrilateral}\:\mathrm{to}\:\mathrm{the}\:\mathrm{opposite} \\ $$$$\mathrm{sides}\:\mathrm{are}\:\mathrm{concurrent}. \\ $$

Question Number 16074 Answers: 0 Comments: 0

Let K, L, M and N be the midpoints of the sides AB, BC, CD and DA, respectively, of a cyclic quadrilateral ABCD. Prove that the orthocenters of the triangles AKN, BKL, CLM and DMN are the vertices of a parallelogram.

$$\mathrm{Let}\:{K},\:{L},\:{M}\:\mathrm{and}\:{N}\:\mathrm{be}\:\mathrm{the}\:\mathrm{midpoints}\:\mathrm{of} \\ $$$$\mathrm{the}\:\mathrm{sides}\:{AB},\:{BC},\:{CD}\:\mathrm{and}\:{DA}, \\ $$$$\mathrm{respectively},\:\mathrm{of}\:\mathrm{a}\:\mathrm{cyclic}\:\mathrm{quadrilateral} \\ $$$${ABCD}.\:\mathrm{Prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{orthocenters} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{triangles}\:{AKN},\:{BKL},\:{CLM}\:\mathrm{and} \\ $$$${DMN}\:\mathrm{are}\:\mathrm{the}\:\mathrm{vertices}\:\mathrm{of}\:\mathrm{a} \\ $$$$\mathrm{parallelogram}. \\ $$

Question Number 16072 Answers: 1 Comments: 3

Let ABCD be a convex quadrilateral. Prove that the orthocenters of the triangles ABC, BCD, CDA and DAB are the vertices of a quadrilateral congruent to ABCD and prove that the centroids of the same triangles are the vertices of a cyclic quadrilateral.

$$\mathrm{Let}\:{ABCD}\:\mathrm{be}\:\mathrm{a}\:\mathrm{convex}\:\mathrm{quadrilateral}. \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{orthocenters}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{triangles}\:{ABC},\:{BCD},\:{CDA}\:\mathrm{and}\:{DAB} \\ $$$$\mathrm{are}\:\mathrm{the}\:\mathrm{vertices}\:\mathrm{of}\:\mathrm{a}\:\mathrm{quadrilateral} \\ $$$$\mathrm{congruent}\:\mathrm{to}\:{ABCD}\:\mathrm{and}\:\mathrm{prove}\:\mathrm{that}\:\mathrm{the} \\ $$$$\mathrm{centroids}\:\mathrm{of}\:\mathrm{the}\:\mathrm{same}\:\mathrm{triangles}\:\mathrm{are}\:\mathrm{the} \\ $$$$\mathrm{vertices}\:\mathrm{of}\:\mathrm{a}\:\mathrm{cyclic}\:\mathrm{quadrilateral}. \\ $$

Question Number 16071 Answers: 1 Comments: 0

Let A′, B′ and C′ be points on the sides BC, CA and AB of the triangle ABC. Prove that the circumcircles of the triangles AB′C′, BA′C′ and CA′B′ have a common point. Prove that the property holds even if the points A′, B′ and C′ are collinear.

$$\mathrm{Let}\:{A}',\:{B}'\:\mathrm{and}\:{C}'\:\mathrm{be}\:\mathrm{points}\:\mathrm{on}\:\mathrm{the}\:\mathrm{sides} \\ $$$${BC},\:{CA}\:\mathrm{and}\:{AB}\:\mathrm{of}\:\mathrm{the}\:\mathrm{triangle}\:{ABC}. \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{circumcircles}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{triangles}\:{AB}'{C}',\:{BA}'{C}'\:\mathrm{and}\:{CA}'{B}' \\ $$$$\mathrm{have}\:\mathrm{a}\:\mathrm{common}\:\mathrm{point}.\:\mathrm{Prove}\:\mathrm{that}\:\mathrm{the} \\ $$$$\mathrm{property}\:\mathrm{holds}\:\mathrm{even}\:\mathrm{if}\:\mathrm{the}\:\mathrm{points}\:{A}', \\ $$$${B}'\:\mathrm{and}\:{C}'\:\mathrm{are}\:\mathrm{collinear}. \\ $$

Question Number 16070 Answers: 0 Comments: 0

Let ABCD be a convex quadrilateral. Prove that AB.CD + AD.BC = AC.BD if and only if ABCD is cyclic (Ptolemy′s theorem).

$$\mathrm{Let}\:{ABCD}\:\mathrm{be}\:\mathrm{a}\:\mathrm{convex}\:\mathrm{quadrilateral}. \\ $$$$\mathrm{Prove}\:\mathrm{that} \\ $$$${AB}.{CD}\:+\:{AD}.{BC}\:=\:{AC}.{BD} \\ $$$$\mathrm{if}\:\mathrm{and}\:\mathrm{only}\:\mathrm{if}\:{ABCD}\:\mathrm{is}\:\mathrm{cyclic}\:\left(\mathrm{Ptolemy}'\mathrm{s}\right. \\ $$$$\left.\mathrm{theorem}\right). \\ $$

Question Number 16069 Answers: 0 Comments: 0

In the interior of a quadrilateral ABCD, consider a variable point P. Prove that if the sum of distances from P to the sides is constant, then ABCD is a parallelogram.

$$\mathrm{In}\:\mathrm{the}\:\mathrm{interior}\:\mathrm{of}\:\mathrm{a}\:\mathrm{quadrilateral} \\ $$$${ABCD},\:\mathrm{consider}\:\mathrm{a}\:\mathrm{variable}\:\mathrm{point}\:{P}. \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{if}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{distances}\:\mathrm{from} \\ $$$${P}\:\mathrm{to}\:\mathrm{the}\:\mathrm{sides}\:\mathrm{is}\:\mathrm{constant},\:\mathrm{then}\:{ABCD} \\ $$$$\mathrm{is}\:\mathrm{a}\:\mathrm{parallelogram}. \\ $$

Pg 105 Pg 106 Pg 107 Pg 108 Pg 109 Pg 110 Pg 111 Pg 112 Pg 113 Pg 114

Contact: info@tinkutara.com