# Euclidean Geometry Grade 12 Notes, Questions and Answers

Euclidean geometry Grade 12 Notes, Questions and Answers: Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid’s method consists in assuming a small set of intuitively appealing axioms and deducing many other propositions from these.

## Euclidean Geometry Grade 12 Questions and Answers (Downloadable pdf)

Here are several Euclidean Geometry questions for Grade 12 students, along with detailed explanations for each answer:

### Question 1: Proving a Triangle is Isosceles

**Prove that triangle ABC is isosceles if AB = AC and angle BAC is bisected by line AD where D is on BC.**

**Answer:** To prove that triangle ABC is isosceles, we can use the properties of isosceles triangles and angle bisectors.

- Given that AB = AC, triangle ABC is isosceles.
- Since AD is the bisector of angle BAC, angle BAD = angle CAD.
- By the definition of angle bisector, it divides the angle into two equal parts.

Hence, triangle ABC is isosceles with AB = AC.

### Question 2: Finding the Length of a Segment

**In triangle ABC, D is the midpoint of BC. If AB = 7 cm, AC = 9 cm, and BC = 10 cm, find the length of AD.**

**Answer:** To find the length of AD, we can use Apollonius’s theorem, which states:

Below you will find a list of Euclidean Geometry Grade 12 Questions and Answers from previous years’ activities, tests, and notes:

### Question 3: Proving Lines are Parallel

**In quadrilateral ABCD, AD is parallel to BC. Prove that angle DAB is equal to angle BCD.**

**Answer:** To prove that angle DAB is equal to angle BCD, we use the properties of parallel lines and corresponding angles.

- Since AD is parallel to BC, and AB is a transversal, angle DAB and angle BCD are alternate interior angles.
- Alternate interior angles are equal when two lines are parallel.

Hence, angle DAB is equal to angle BCD.

### Question 4: Calculating Angles in a Circle

**In circle O, chords AB and CD intersect at E inside the circle. If angle AEC = 40 degrees and angle BED = 60 degrees, find angle BEC.**

**Answer:** To find angle BEC, we use the property that the sum of opposite angles formed by two intersecting chords in a circle is 180 degrees.

### Question 5: Area of a Triangle Using Sine Rule

**In triangle PQR, angle PQR is 60 degrees, PQ = 8 cm, and PR = 10 cm. Calculate the area of triangle PQR.**

**Answer:** To find the area of triangle PQR, we can use the formula:

### Question 6: Proving a Line is a Perpendicular Bisector

**Prove that line segment AD is the perpendicular bisector of BC in triangle ABC, where AB = AC and D is the midpoint of BC.**

**Answer:** To prove AD is the perpendicular bisector of BC, we need to show that AD is both perpendicular to BC and bisects BC.

- Since D is the midpoint of BC, BD = DC.
- Since AB = AC, triangle ABD is congruent to triangle ACD by the Side-Angle-Side (SAS) postulate (AB = AC, AD = AD, and angle BAD = angle CAD).
- Because the triangles are congruent, angle ADB = angle ADC = 90 degrees.

Thus, AD is the perpendicular bisector of BC.

#### Watch: Grade 12 Geometry Lesson 1| Euclidean Geometry: Introduction (Triangles and parallel lines)

## Types of triangles

### Types of triangles (EMCHZ)

Name | Diagram | Properties |

Scalene | All sides and angles are different. | |

Isosceles | Two sides are equal in length. The angles opposite the equal sides are also equal. | |

Equilateral | All three sides are equal in length and all three angles are equal. | |

Acute-angled | Each of the three interior angles is less than 9090°. | |

Obtuse-angled | One interior angle is greater than 9090°. | |

Right-angled | One interior angle is 9090°. |

### The mid-point theorem (EMCJ7)

The line joining the mid-points of two sides of a triangle is parallel to the third side and equal to half the length of the third side.