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.
- 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.