Geometry for RRB Group D

Most geometry questions in RRB Group D use one fact. There are no long proofs and no constructions. If you know the fact, the question takes 20 seconds. If you don't, no amount of calculation will save you.

So geometry is a memory topic first and an arithmetic topic second. This guide lists the facts by chapter, shows how each one is asked, and gives you a practice set. Draw a quick rough figure for every question, even when one is given. It stops you from mixing up which angle is which.

Angles and parallel lines

• Angles on a straight line add up to 180°.
• Angles around a point add up to 360°.
• Complementary angles add up to 90°. Supplementary angles add up to 180°.
• Vertically opposite angles are equal.

When a line (a transversal) cuts two parallel lines:

Triangles

• The three angles add up to 180°.
• An exterior angle equals the sum of the two opposite interior angles.
• In an isosceles triangle, the angles opposite the equal sides are equal. In an equilateral triangle, every angle is 60°.
• Any two sides together are longer than the third side. So 3 cm, 4 cm and 8 cm cannot make a triangle.
• The centroid (where the medians meet) divides each median in the ratio 2 : 1 from the vertex.

Pythagoras and triplets

In a right-angled triangle, (hypotenuse)² = (base)² + (height)². The hypotenuse is always the side opposite the right angle and always the longest side.

Learn these triplets, because the setters use them again and again:

Two special right triangles also appear, and they link to trigonometry:

• 30°–60°–90°: sides in the ratio 1 : √3 : 2 (the shortest side faces 30°).
• 45°–45°–90°: sides in the ratio 1 : 1 : √2.

Polygons

For a polygon with n sides:

Circles

• The angle in a semicircle is 90°.
• The angle at the centre is twice the angle at the circumference standing on the same arc.
• Angles in the same segment are equal.
• Opposite angles of a cyclic quadrilateral add up to 180°.
• A tangent meets the radius at 90° at the point of contact.
• Two tangents drawn from the same outside point are equal in length.
• The perpendicular from the centre to a chord bisects the chord.

The last three facts turn many circle questions into Pythagoras questions.

Worked examples

Example 1: The angles of a triangle are in the ratio 2 : 3 : 4. Find them.

• Total parts = 2 + 3 + 4 = 9. One part = 180 ÷ 9 = 20°.
• Angles: 40°, 60° and 80°.

Example 2: A right triangle has sides 9 cm and 12 cm around the right angle. Find the hypotenuse.

• 9, 12 is 3 × (3, 4). So the hypotenuse is 3 × 5 = 15 cm.
• Check: 81 + 144 = 225 = 15².

Example 3: Two parallel lines are cut by a transversal. One co-interior angle is 65°. Find the other.

• Co-interior angles add up to 180°. The other angle = 180 − 65 = 115°.

Example 4: Each exterior angle of a regular polygon is 45°. How many sides does it have, and how many diagonals?

• Sides = 360 ÷ 45 = 8.
• Diagonals = 8 × (8 − 3) ÷ 2 = 8 × 5 ÷ 2 = 20.

Example 5: A chord of a circle of radius 13 cm is 24 cm long. How far is it from the centre?

• The perpendicular from the centre bisects the chord, so half the chord is 12 cm.
• Distance = √(13² − 12²) = √(169 − 144) = √25 = 5 cm. (5, 12, 13 again.)

Example 6: A point is 17 cm from the centre of a circle of radius 8 cm. Find the length of the tangent from the point.

• The tangent meets the radius at 90°, so the distance to the centre is the hypotenuse.
• Tangent = √(17² − 8²) = √(289 − 64) = √225 = 15 cm.

Common mistakes

• Using Pythagoras on a triangle that has no right angle.
• Taking the longest given number as a side instead of the hypotenuse, or the other way round.
• Mixing up complement (90°) and supplement (180°).
• Using (n − 2) × 180° as the size of each angle, when it is the sum of all the interior angles.
• Forgetting to halve the chord before using Pythagoras.

Practice set

1. Find the complement of 58°.
2. A right triangle has hypotenuse 25 cm and one side 7 cm. Find the other side.
3. Find each interior angle of a regular hexagon.
4. Each exterior angle of a regular polygon is 30°. How many sides does it have?
5. An exterior angle of a triangle is 110°. One of the opposite interior angles is 50°. Find the other.
6. The vertex angle of an isosceles triangle is 40°. Find each base angle.
7. One angle of a cyclic quadrilateral is 75°. Find the angle opposite to it.
8. An arc makes an angle of 130° at the centre of a circle. What angle does it make at a point on the remaining part of the circle?

Answers:

1. 32°. 90 − 58.
2. 24 cm. √(625 − 49) = √576. This is the 7, 24, 25 triplet.
3. 120°. Each exterior angle is 360 ÷ 6 = 60°, and 180 − 60 = 120.
4. 12. 360 ÷ 30.
5. 60°. The exterior angle equals the sum of the opposite interior angles: 110 − 50.
6. 70° each. (180 − 40) ÷ 2.
7. 105°. Opposite angles of a cyclic quadrilateral add up to 180°.
8. 65°. The angle at the circumference is half the angle at the centre.

What to do next

• Copy the triplet table and the circle facts onto one card and read it daily for a week.
• For every geometry question you practise, draw the rough figure first.
• Solve 20 mixed questions, then note which fact each wrong answer needed.
• Carry these shapes into mensuration, where you find their areas and volumes.