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Lines & Triangles
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CAT 2025 Lesson : Lines & Triangles - Concepts & Cheatsheet

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Note: The video for this module contains a summary of all the concepts covered in this lesson. The video would serve as a good revision. Please watch this video in intervals of a few weeks so that you do not forget the concepts. Below is a cheatsheet that includes all the formulae but not necessarily the concepts covered in the video.

7. Cheatsheet

1) Sum of any 22 sides of a triangle is always greater than the third side.

2) Number of triangles with integral sides, where the perimeter pp is even == p248\dfrac{p^{2}}{48} and pp is odd == (p+3)248\dfrac{(p + 3)^{2}}{48} rounded to the nearest integer.

3) Sum of the interior and exterior angles of a triangle are 180o180^\mathrm{o} and 360o360^\mathrm{o} respectively.

4) Exterior angle of a triangle == Sum of its opposite interior angles

5) Where aa is the longest side of a triangle, in a(n)
- Acute Angled Triangle, a2<b2+c2a^{2} < b^{2} + c^{2}
- Obtuse Angled Triangle, a2>b2+c2a^{2} > b^{2} + c^{2}
- Right Angled Triangle, a2=b2+c2a^{2} = b^{2} + c^{2}

6) In △\triangleABC, if the altitude AD lies
- inside the triangle, then AC2^{2} == AB2^{2} + BC2^{2} −(2×- (2 \times BD ×\times BC)
- outside the triangle, then AC2^{2} == AB2^{2} + BC2^{2} +(2×+ (2 \times BD ×\times BC)

7) The points where the altitudes meet is called the orthocentre, where the perpendicular bisectors meet is called the circumcentre, where the angle bisectors meet is called the incentre and medians meet is called the centroid. (Refer Module 5 for their properties)

8) Area of Triangle == 12bh\dfrac{1}{2}bh
                                  == s(s−a)(s−b)(s−c)\sqrt{s (s - a) (s - b) (s - c)}
                                  == 12absinθ\dfrac{1}{2}ab sin \theta
                                  == r×s=abc4Rr \times s = \dfrac{abc}{4R}

9) Sine Rule: asinA=bsinB=\dfrac{a}{sinA} = \dfrac{b}{sin B} = csinC\dfrac{c}{sin C} and Cosine Rule: cos A = b2+c2−a22bc\dfrac{b^{2} + c^{2} - a^{2}}{2bc}

11) In an isosceles triangle, Area == b44a2−b2\dfrac{b}{4}\sqrt{4a^{2} - b^{2}}, where aa is the length of the equal sides.

12) In an equilateral triangle where aa is the length of a side, Area == 34a2\dfrac{\sqrt{3}}{4}a^{2}, Height == hh == 32a\dfrac{\sqrt{3}}{2}a, Inradius == 13×h=\dfrac{1}{3} \times h = a23\dfrac{a}{2\sqrt{3}} , Circumradius: 23×h=\dfrac{2}{3} \times h = a3\dfrac{a}{\sqrt{3}}

13) In Right-angled triangles △\triangleABC, where AC is the hypotenuse,
Pythagoras Theorem: AB2^{2} + BC2^{2} == AC2^{2} ; Area of △\triangleABC == 12×AB×BC\dfrac{1}{2} \times AB \times BC

Circumradius == AC2\dfrac{AC}{2} ; Inradius == AB+BC−AC2\dfrac{AB + BC - AC}{2}

(Refer Module 8 for other properties)

14) Where the angles of a right-angled triangle are
- 30o−60o−90o30^\mathrm{o} - 60^\mathrm{o} - 90^\mathrm{o}, then sides opposite them are in the ratio 1:3:21 : \sqrt{3} : 2.
- 45o−45o−90o45^\mathrm{o} - 45^\mathrm{o} - 90^\mathrm{o}, then sides opposite them are in the ratio 1:1:21 : 1 : \sqrt{2}.

15) Apollonius Theorem: In △\triangleABC, where AD is the median, AB2^{2} ++ AC2^{2} == 22(AD2^{2} ++ BD2^{2})

16) Extension of Apollonius Theorem: In △\triangleABC, where AD, BE and CF are the medians, AD2^{2} ++ BE2^{2} ++ CF2^{2} == 34\dfrac{3}{4}(AB2^{2} ++ BC2^{2} ++ AC2^{2})

17) Interior Angle Bisector Theorem: In △\triangleABC, where AD is the interior angle bisector, ABAC=BDCD\dfrac{AB}{AC} = \dfrac{BD}{CD}

18) Exterior Angle Bisector Theorem: In △\triangleABC, where AD is the external angle bisector of ∠\angleA and D is a point on extended BC, ABAC=BDCD\dfrac{AB}{AC} = \dfrac{BD}{CD}

19) Basic Proportionality Theorem: A line drawn parallel to one of the sides cuts the other 22 sides in the same ratio, i.e. in △\triangleABC, if DE | | BC, then ADDB=AEEC\dfrac{AD}{DB} = \dfrac{AE}{EC}

20) Midpoint Theorem: The line joining the mid-points of any 22 sides of a triangle will be parallel to the third side.
Likewise, if a line bisects one side and is parallel to another side of a triangle, then it bisects the third side.

21) Congruency Rules are SSS, SAS, ASA and RHS.

22) Similarity Rules are SSS, SAS and AA.

23) When △\triangleABC ~ △\triangleDEF, the following are to be noted.
- ABDE=BCEF=CAFD\dfrac{AB}{DE} = \dfrac{BC}{EF} = \dfrac{CA}{FD}

- Ratio of Sides == Ratio of Perimeters == Ratio of Altitudes == Ratio of Medians == Ratio of Angle Bisectors = Ratio of Perpendicular Bisectors == Ratio of Inradii == Ratio of Circumradii
area(△ABC)area(△DEF)=\dfrac{area( \triangle{ABC})}{area( \triangle{DEF})} = AB2DE2=BC2EF2\dfrac{AB^{2}}{DE^{2}} = \dfrac{BC^{2}}{EF^{2}} =CA2FD2= \dfrac{CA^{2}}{FD^{2}}

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