Triangles: SSC CGL Quantitative Aptitude

Formulas (3)

• Angle sum of a triangle, with ratios and equations · Angle sum
  $$\angle A + \angle B + \angle C = 180^\circ \qquad \text{one part} = \frac{180^\circ}{a + b + c}$$
• The exterior angle of a triangle · Exterior angle
  $$\angle ACD = \angle A + \angle B = 180^\circ - \angle ACB$$
• Isosceles and equilateral triangles · Base angle of an isosceles triangle
  $$\text{base angle} = \frac{180^\circ - \text{apex}}{2}$$

Watch out for (7)

• Dividing 180° by one part of the ratio
  → Angle sum of a triangle, with ratios and equations
• Allowing two obtuse or right angles
  → Angle sum of a triangle, with ratios and equations
• Giving the difference in parts, not degrees
  → Angle sum of a triangle, with ratios and equations
• Adding the interior angle beside the exterior one
  → The exterior angle of a triangle
• Answering with the exterior angle as an interior one
  → The exterior angle of a triangle
• Putting the equal angles at the wrong corners
  → Isosceles and equilateral triangles
• Halving the apex to get a base angle
  → Isosceles and equilateral triangles

Formulas (3)

• Triangle inequality: the third side lies between the difference and the sum · Range of the third side
  $$|a - b| < x < a + b \qquad \text{each side} < \frac{a + b + c}{2}$$
• Pythagoras theorem, triples and the acute or obtuse test · Pythagoras and the equilateral height
  $$c^2 = a^2 + b^2 \qquad h = \frac{\sqrt{3}}{2}a \qquad PQ^2 - QS^2 = PR^2 - SR^2$$
• The altitude to the hypotenuse of a right triangle · Altitude to the hypotenuse
  $$BD = \frac{AB \times BC}{AC} \qquad AB^2 = AD \times AC \qquad BD^2 = AD \times DC$$

Watch out for (7)

• Including the ends of the range
  → Triangle inequality: the third side lies between the difference and the sum
• Counting from the difference itself
  → Triangle inequality: the third side lies between the difference and the sum
• Repeating the short side of an isosceles triangle
  → Triangle inequality: the third side lies between the difference and the sum
• Testing with a side that is not the longest
  → Pythagoras theorem, triples and the acute or obtuse test
• Using the whole base in an isosceles triangle
  → Pythagoras theorem, triples and the acute or obtuse test
• Taking the radius as the hypotenuse
  → The altitude to the hypotenuse of a right triangle
• Pairing a leg with the far piece
  → The altitude to the hypotenuse of a right triangle

Formulas (1)

• Reading the matching corners from a congruence statement · Matching parts
  $$\triangle ABC \cong \triangle PQR \;\Rightarrow\; AB = PQ,\; BC = QR,\; CA = RP,\; \angle A = \angle P,\; \angle B = \angle Q,\; \angle C = \angle R$$

Watch out for (7)

• Taking equal angles as congruence
  → What congruent triangles have in common
• Believing congruent triangles can differ in area
  → What congruent triangles have in common
• Matching corners by alphabet or by position
  → Reading the matching corners from a congruence statement
• Trusting how the figure is drawn
  → Reading the matching corners from a congruence statement
• Accepting two sides and a non-included angle
  → The congruence rules: SSS, SAS, ASA and RHS
• Calling a right-angle case SAS
  → The congruence rules: SSS, SAS, ASA and RHS
• Missing the shared side or the crossing angles
  → The congruence rules: SSS, SAS, ASA and RHS

Formulas (3)

• Similar triangles and the scale factor for lengths · Scale factor of similar triangles
  $$k = \frac{AB}{PQ} = \frac{BC}{QR} = \frac{CA}{RP} = \frac{\text{perimeter of } ABC}{\text{perimeter of } PQR}$$
• Areas of similar triangles: the square of the scale factor · Areas of similar figures
  $$\frac{\text{Area}_1}{\text{Area}_2} = k^2 = \left(\frac{\text{side}_1}{\text{side}_2}\right)^2$$
• A line parallel to one side: the areas it cuts off · Area cut off by a parallel line
  $$\frac{\text{ar}(ADE)}{\text{ar}(ABC)} = \left(\frac{AD}{AB}\right)^2 = \left(\frac{DE}{BC}\right)^2$$

Watch out for (8)

• Reading similar as equal
  → Similar triangles and the scale factor for lengths
• SAS similarity with the wrong angle
  → Similar triangles and the scale factor for lengths
• Turning the ratio over halfway
  → Similar triangles and the scale factor for lengths
• Using k for areas
  → Areas of similar triangles: the square of the scale factor
• Squaring on the way back
  → Areas of similar triangles: the square of the scale factor
• Squaring the perimeter ratio for perimeters
  → Areas of similar triangles: the square of the scale factor
• Squaring AD : DB
  → A line parallel to one side: the areas it cuts off
• Treating the strip as the whole triangle
  → A line parallel to one side: the areas it cuts off

Formulas (3)

• Basic proportionality theorem: a line parallel to one side · Basic proportionality theorem
  $$\frac{AD}{DB} = \frac{AE}{EC} \qquad \frac{DE}{BC} = \frac{AD}{AB}$$
• The midpoint theorem and the quarter triangle · Midpoint theorem
  $$XY = \frac{1}{2}BC \qquad \text{ar}(XYZ) = \frac{1}{4}\,\text{ar}(ABC)$$
• The angle bisector theorem · Angle bisector theorem
  $$\frac{BD}{DC} = \frac{AB}{AC} \qquad BD = \frac{AB}{AB + AC} \times BC$$

Watch out for (6)

• Using AD : DB for the parallel segment
  → Basic proportionality theorem: a line parallel to one side
• Reading 5AE = 3EC as AE : EC = 5 : 3
  → Basic proportionality theorem: a line parallel to one side
• Taking the midpoint triangle as half the area
  → The midpoint theorem and the quarter triangle
• Calling the midpoint triangle congruent to the whole
  → The midpoint theorem and the quarter triangle
• Pairing each piece with the far side
  → The angle bisector theorem
• Putting the third side into the ratio
  → The angle bisector theorem

Formulas (3)

• The centroid divides each median in the ratio 2 : 1 · Centroid on a median
  $$AG = \frac{2}{3}AD \qquad GD = \frac{1}{3}AD$$
• A median halves the area, and the centroid makes equal parts · Areas made by medians
  $$\text{ar}(ABD) = \frac{1}{2}\,\text{ar}(ABC) \qquad \text{ar}(GBC) = \frac{1}{3}\,\text{ar}(ABC)$$
• Apollonius theorem: the length of a median · Apollonius theorem
  $$AB^2 + AC^2 = 2(AD^2 + BD^2) \qquad AD = \frac{1}{2}\sqrt{2AB^2 + 2AC^2 - BC^2}$$

Watch out for (6)

• Reading 2 : 1 from the wrong end
  → The centroid divides each median in the ratio 2 : 1
• Using whole medians when two medians cross at 90°
  → The centroid divides each median in the ratio 2 : 1
• Taking the centroid triangles as quarters
  → A median halves the area, and the centroid makes equal parts
• Thinking a median quarters the area
  → A median halves the area, and the centroid makes equal parts
• Dropping the 2 in Apollonius
  → Apollonius theorem: the length of a median
• Using the whole side where the half is needed
  → Apollonius theorem: the length of a median

Formulas (3)

• The angle at the incentre between two angle bisectors · Angle at the incentre
  $$\angle BIC = 90^\circ + \frac{\angle A}{2}$$
• Inradius: the area divided by the semi-perimeter · Inradius
  $$r = \frac{\text{Area}}{s} \qquad s = \frac{a + b + c}{2} \qquad r_{\text{right}} = \frac{a + b - c}{2}$$
• Circumradius of a right, an equilateral and any triangle · Circumradius
  $$R = \frac{abc}{4 \times \text{Area}} \qquad R_{\text{right}} = \frac{\text{hypotenuse}}{2} \qquad R_{\text{equilateral}} = \frac{a}{\sqrt{3}}$$

Reference tables (1)

Watch out for (8)

• Swapping the circumcentre and the orthocentre of a right triangle
  → The four centres of a triangle and where each one lies
• Making the incentre equidistant from the corners
  → The four centres of a triangle and where each one lies
• Adding the whole of the opposite angle
  → The angle at the incentre between two angle bisectors
• Using the wrong angle of the triangle
  → The angle at the incentre between two angle bisectors
• Dividing the area by the full perimeter
  → Inradius: the area divided by the semi-perimeter
• Adding the hypotenuse in the right-triangle form
  → Inradius: the area divided by the semi-perimeter
• Halving a leg instead of the hypotenuse
  → Circumradius of a right, an equilateral and any triangle
• Swapping R and r in an equilateral triangle
  → Circumradius of a right, an equilateral and any triangle