Draw ΔABC with AB = 5 cm, ∠A = 70° and ∠B = 60°. Draw the circumcircle of ΔABC. Is the centre inside or outside the triangle?
Step-by-step solution
To find: The circumcircle of ΔABC, and whether its centre is inside or outside the triangle
Idea: The circumcentre is the point where the perpendicular bisectors of the sides meet; it is equidistant from A, B and C. In an acute-angled triangle it lies inside, in an obtuse-angled triangle outside, and in a right-angled triangle on the hypotenuse.
- Draw the triangle: draw AB = 5 cm. At A construct a ray making 70° with AB, and at B a ray making 60° with BA, on the same side. The rays meet at C.1 mark
- Perpendicular bisectors: with compasses, draw the perpendicular bisectors of AB and of BC. Mark their point of intersection O.½ mark
- Circumcircle: with centre O and radius OA, draw the circle. It passes through A, B and C (OA = OB = OC ≈ 3.3 cm).½ mark
- ∠C = 180° − 70° − 60° = 50°. All three angles (70°, 60°, 50°) are less than 90°, so ΔABC is acute-angled.½ mark
- The figure shows O inside ΔABC, as expected for an acute-angled triangle. The centre is inside the triangle.½ mark
Check: Measure OA, OB and OC: each should be about 3.3 cm. (By calculation the radius is 5 ÷ (2 sin 50°) ≈ 3.26 cm.)
Answer to write in the exam
Steps of construction:
1. Draw AB = 5 cm.
2. At A, draw ∠BAX = 70°; at B, draw ∠ABY = 60°. AX and BY meet at C.
3. Draw the perpendicular bisectors of AB and BC; they meet at O.
4. With centre O and radius OA, draw a circle. It passes through A, B and C.
∠C = 180° − 70° − 60° = 50° (angle sum of a triangle) ⇒ ΔABC is acute-angled
∴ The centre O lies inside the triangle. (OA = OB = OC ≈ 3.3 cm)
Common mistakes that cost marks
- Constructing 60° at B on the wrong side of AB, so the two rays never meet to form the triangle.
- Drawing angle bisectors instead of perpendicular bisectors. Angle bisectors give the centre of the circle inside the triangle touching its sides, not the circle through the vertices.
- Using a radius other than OA. The circle must pass through all three vertices; check that it does.
How this can come in the exam
In ΔPQR, ∠P = 50° and ∠Q = 60°. Where does the circumcentre of ΔPQR lie?
- Inside the triangle
- Outside the triangle
- At the midpoint of PR
- At the vertex R
Show answer
(A) Inside the triangle
∠R = 180° − 50° − 60° = 70°. All angles are acute, so the circumcentre lies inside the triangle.
Try one yourself
Draw ΔPQR with QR = 6 cm, ∠Q = 45° and ∠R = 75°. Draw its circumcircle. Is the centre inside or outside?
Show answer
∠P = 180° − 45° − 75° = 60°. All angles are acute, so the circumcentre is inside. (Circumradius = 6 ÷ (2 sin 60°) ≈ 3.5 cm.)
More questions like this
- Draw ΔABC with AB = 5 cm, ∠A = 100°, AC = 4 cm. Draw the circumcircle of ΔABC. Is the centre inside or outside the triangle?
- Draw ΔABC, with AB = 6 cm, BC = 7 cm and CA = 7 cm. Draw the circumcircle of ΔABC. Let the circumcentre be O. Measure OA, OB, OC.
- What is the least possible radius of a circle through two points A and B?
- 1. A, B and C are three collinear points. Can you find a point P such that PA = PB = PC ? What can you say about the perpendicular bisectors of AB and BC? Draw and check. Can you show that for three collinear points A, B and C, the perpendicular bisector of AB and BC are parallel? Is it possible for a circle to pass through collinear points? Can you draw a line that cuts a given circle in three distinct points?
2. The circumcircle of a given ΔABC is drawn. Can there be other triangles congruent to ΔABC that share the same circumcircle? - Equal chords of a circle subtend equal angles at the centre of the circle.