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.

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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.

  1. Given that AB = AC, triangle ABC is isosceles.
  2. Since AD is the bisector of angle BAC, angle BAD = angle CAD.
  3. 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:

Finding the Length of a Segment

Question 3: Proving Lines are Parallel

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

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

  1. Since AD is parallel to BC, and AB is a transversal, angle DAB and angle BCD are alternate interior angles.
  2. 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.

Calculating Angles in a Circle

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:

Area of a Triangle Using Sine Rule

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.

  1. Since D is the midpoint of BC, BD = DC.
  2. 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).
  3. Because the triangles are congruent, angle ADB = angle ADC = 90 degrees.
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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)

NameDiagramProperties
Scalene009059e85d03a14b0535ccbe43991bb0.pngAll sides and angles are different.
Isosceles4e56e4dc5efd0035fdbf310a383e261f.pngTwo sides are equal in length. The angles opposite the equal sides are also equal.
Equilateralequilateral.pngAll three sides are equal in length and all three angles are equal.
Acute-angledcc9650766b7bf6798ac6a86ea8cd3cb4.pngEach of the three interior angles is less than 9090°.
Obtuse-angled501a63891137afb1e0ba73111f218a87.pngOne interior angle is greater than 9090°.
Right-angledd3102e4ac20d4af698e91bd839b26039.pngOne 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.91b343e1387d3cac099f2ab8ddb74a2f.png