Author’s Solution for Challenging Problems in Circles
1. Question:
AB & CD are two chords of a circle with centre ‘O’ intersecting each other at right angles. Prove that ∠AOC + ∠BOD = 180
[ Here, is an interesting story behind the problem. This problem was given to our author as an assignment when he was at his SSLC. The teacher has given instruction to do this problem using the theorem ‘The angle made by a chord at the centre is twice the angle at the circumference’ but instead our author has solved this problem in the other way which is beyond his level. Though the teacher initially got angry on looking at the solution (as our author didn’t apply the theorem taught to him), at last he appreciated our author for solving the problem the other way though it is complicated.]
The Teacher’s expected solution is also given below.
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- Parthiban v
30-Dec-2022
2. Question:
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- Parthiban v
30-Dec-2022
First test comment, from the Vendor. Have a nice day!!
-
30-Dec-2022
3. Question:
If the side of the regular hexagon is 6cm, the find the radius of the circle
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- Parthiban v
30-Dec-2022
4. Question:
Find the radius of the circle if AE = 3cm, CG = 5cm, AE ⊥ CE & CE ⊥ CG.
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- Parthiban v
30-Dec-2022
5. Question:
Δ APB is a right angled triangle & O is the circumcentre of the triangle and a tangents drawn at P. From A & B perpendiculars are drawn to meet the tangent at D & C respectively.
Prove that AD + BC = AB
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- Parthiban v
30-Dec-2022
6. Question:
ABC be an acute angled triangle and O be its circumcentre. A circle through the points A, O and B is drawn and CA produced and CB meets the circle at P, Q respectively. Prove that CO & PQ are perpendicular to each other.
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- Parthiban v
30-Dec-2022
7. Question:
ABC is a triangle inscribed in a circle. M is the mid point of major arc AB and D is the foot of the perpendicular from M on AC. Then prove that AD = BC + CD.
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- Parthiban v
30-Dec-2022
8. Question:
A square ABCD and an equilateral triangle PQR are inscribed in a circle centered at O in such a way that AB is parallel to QR. The sides PQ and PR of the triangle meet the side AB of the square at X and Y respectively. Find the value of ∠XOY.
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- Parthiban v
30-Dec-2022
9. Question:
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30-Dec-2022
Author’s Solution for Challenging Problems in Triangles
1. Question:
In Δ ABC, ∠B — ∠C =x°. O is the circumcenter and ‘D’ is the midpoint of BC. Prove that ∠AOD = 180° — x°
Question created by
Dr.M.Raja Climax
Founder Chairman, CEOA
2. Question:
In Δ ABC, AC=BC and ∠C=20°. If D & E are points on AC & BC such that ∠EAB = 70° & ∠DBA = 60°, then find ∠DEA.
For this problem, the author has given two different solutions one using geometry and the other using trigonometry.
3. Question:
Give non-collinear points A,B,C, segment AB is trisected by points D & E, and F is the mid-point of segment AC. DF & BF intersect CE at G & H respectively.
If area of Δ DEG is 18 Sq.units, find the area of Δ FGH.
4. Question:
Find the area of the shaded region
5. Question:
For this above Problem, You can find some lengthy & complicated solutions which is hard to follow in the net but our Author has given not one or two but three various Simple Solutions which can be viewed below.
6. Question:
In Δ ABC, find the area of the shaded area to unshaded area if AD=DE=EC & BF=CF.
7. Question:
In Δ ABC, ∠A = 100º & AB = AC. The angle bisector of ∠B meets AC at D. Prove that BD + AD =BC
8. Question:
In ΔABC, AD is a median from A to BC such that ∠DAC=15° and ∠ADB=45º, find x
9. Author's Corollary 1
Corollary If AD is a median from A to BC in such that ∠ADB=45º and BE is drawn perpendicular to AD meeting AC at F, then tan θ = AF/AC
10. Author's Corollary 2
If AD is a median from A to BC in such that ∠ADB=45° and ∠DAC=15° .BE is drawn perpendicular to AD meeting AC at F, then AD x BF = 2 BD²
12. Question
In triangle ABC, ∠B = 2∠C, if AD is the angle bisector such that DC = AB, then find angle A
13. Question
ABC is a triangle right angled at C. BC is divided by points D & E into three equal parts. Find the sum of the angles AEC, ADC & ABC if BC = 3AC.
14. Question
∆ ABC is an equilateral triangle. D is a point outside ∆ABC such that
BD=CD=a cm and ∠𝐵𝐷𝐶=120°.
M, N are points on AB, AC respectively such that ∠𝑀𝐷𝑁=60°.
Prove that MN+AN+AM = AB+AC.
Corollary
- 1. Triangles MDN, MBP, QDP & QCN are similar to each other.
- 2. ∠ANM = 2∠CDN
- 3. Triangles MQN & DCN are similar
- 4. Triangles MBD & MPN are similar
- 5. MNQP is concyclic
15. Question
In, ΔABC, ∠ABC = 45º, AD is a cevian meeting BC at D such that DC = 2BD and ∠ADC = 60º, find ∠ACB.
16. Question
In ∆ 𝑨𝑩𝑪,𝑨𝑩 = 𝟗, 𝑨𝑪 = 𝟏𝟎 𝒂𝒏𝒅 𝑩𝑪 = 𝟏𝟏.Point D is on AC such that BD is an angle bisector, and E is on BD such that CE ⊥ 𝑩𝑫.Compute the area of ∆ 𝑨𝑩𝑬.
17. Author's Corollary
ABCis a Δ. BD is the bisector of ∠B meeting AC at D. CEis drawn ⊥ BD or BD produced. Prove that Area of Δ ABE = 1/2 area of Δ ABC
18. Question
In triangle ΔABC, AB = AC and A = ∠120° . D,E are trisection points of BC. Prove that ΔADE is equilateral.
Author’s Solution for Challenging Problems in Quadrilaterals
1. Question:
ABCD is a quadrilateral. Its diagonals AC & BD measuring 6 cm & 5 cm respectively cut each other at O and ∠AOD is 30°. Find the area of the quadrilateral ABCD.
Question created by
Dr.M.Raja Climax
Founder Chairman, CEOA
2. Question:
Find the value of θ, if ∠PAB = ∠ PBA = 15º
The question sender was taken aback on viewing two simple solutions for such a challenging problem.
3. Question:
Find tanθ in the given figure
4. Question:
P is a point on side AB of square ABCD such that DP =5cm. DQ is the angle bisector of ∠PDC where Q is a point on side BC. Then find the length of (CQ+AP)
5. Question:
If ABCD is a square with side 5 cm, E, F, G are the midpoints of the sides AB, BC, CD respectively. Find the area of the shaded region
6. Question:
ABCD is a cyclic quadrilateral. AC & BD meet at P. O is the circumcentre of ∆ APB. Prove that, PZ is an altitude of ∆ CPD and there by prove that O, P and the orthocentre of ∆CPD are collinear,
7. Question:
In the square ABCD, E is the interior point such that ∠AED is 90º. F is point on DE such that ∠CFD is 90º. AF meets CD at G. CE meets DA at H. Lines GH and CF meets at N, GH and AE meet at M. Then prove that GN=HM
8. Question:
ABCD is a square. E & F are mid-points of sides AD & BC respectively. Line segments AC,BD,CE & DF intersect in the interior of the square to form a quadrilateral PQRS as shown in the figure. What is the ratio of area PQRS to that of ABCD?
Author’s Solution for Challenging Problems in Mensuration
1. Question:
In the figure, ABCD is a rectangle. AB = 28 units & BC = 14√3 units. Semicircles have been using AB & CD as diameters, as shown in the picture. Find the area of the shaded portion.
