Congruent triangles · Ages 13–14
Bisecting an angle and a line segment
Two arcs of one radius; the line through where they cross bisects.
How to do it
Method
Two arcs of one radius; the line through where they cross bisects
OP = OQ, PR = QR, OR = OR: △OPR ≅ △OQR (SSS)
Another way
A rhombus's diagonals
AP = BP = AQ = BQ: PQ bisects AB at right angles
In full
To bisect ∠AOB: with centre O draw an arc cutting OA at P and OB at Q; with centres P and Q and one radius, draw two arcs crossing at R; join OR, the bisector. Why? OP = OQ, PR = QR and OR is a common side, so △OPR ≅ △OQR (SSS) and ∠POR = ∠QOR. For the perpendicular bisector of AB: with centres A and B and one radius (more than half of AB), draw arcs crossing on either side of AB at P and Q; join PQ. PQ is at right angles to AB and passes through its midpoint M. Every point on the perpendicular bisector is as far from A as from B. Another way to see it: AP = BP = AQ = BQ, so APBQ is a rhombus, and a rhombus's diagonals bisect each other at right angles.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
PQ is the perpendicular bisector of AB, cutting it at M, as shown. AB = 10 cm, AM =
Show the answer and working
Answer: 5 cm
The perpendicular bisector passes through AB's midpoint: AM = 10 ÷ 2 = 5 cm.
Example 2
P is on the perpendicular bisector of the line segment AB, PA = 9 cm and AB = 3 cm. PB =
Show the answer and working
Answer: 9 cm
Every point on the perpendicular bisector is as far from A as from B: PB = PA = 9 cm; AB's length isn't needed.
Example 3
OR is drawn with compasses and a straight edge, as shown (OP = OQ, PR = QR). Why does OR bisect ∠AOB?
- A. OP = OQ, ∠POR = ∠QOR, OR = OR, so △OPR ≅ △OQR (SAS)
- B. OP = OQ, PR = QR, OR = OR, so △OPR ≅ △OQR (SSS)
- C. OP = OQ, PR = QR, so △OPR ≅ △OQR (RHS)
- D. ∠OPR = ∠OQR, OP = OQ, so △OPR ≅ △OQR (ASA)
Choose the right answer
Show the answer and working
Answer: B. OP = OQ, PR = QR, OR = OR, so △OPR ≅ △OQR (SSS)
Three pairs of equal sides (OR common), so △OPR ≅ △OQR (SSS) and ∠POR = ∠QOR (corr. ∠s, ≅△s).
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.Bisect ∠AOB (68°) with compasses and a straight edge: put the steps in order.
- A. ∠AOR = ∠BOR = 34°: join OR
- B. PR = QR: with centres P and Q and one radius, draw arcs crossing at R
- C. OP = OQ: with centre O, draw an arc cutting OA at P and OB at Q
Write the letters in the right order
Answer:
2.Draw the perpendicular bisector of the line segment AB (14 cm) with compasses and a straight edge: put the steps in order.
- A. AQ = BQ: with the same radius, draw arcs again below AB, crossing at Q
- B. AP = BP: with centres A and B and one radius (more than 7 cm), draw arcs crossing above AB at P
- C. AM = MB = 7 cm: join PQ, cutting AB at M
Write the letters in the right order
Answer:
3.OR is drawn with compasses and a straight edge, as shown (OP = OQ, PR = QR). Why does OR bisect ∠AOB?
- A. OP = OQ, ∠POR = ∠QOR, OR = OR, so △OPR ≅ △OQR (SAS)
- B. OP = OQ, PR = QR, OR = OR, so △OPR ≅ △OQR (SSS)
- C. OP = OQ, PR = QR, so △OPR ≅ △OQR (RHS)
- D. ∠OPR = ∠OQR, OP = OQ, so △OPR ≅ △OQR (ASA)
Choose the right answer
Answer:
4.OR bisects ∠AOB, as shown, and ∠AOB = 74°. ∠AOR =
Answer:
5.The perpendicular bisector PQ of AB cuts it at M, as shown; AB = 18 cm and PA = 15 cm. PB = cm, MB = cm, ∠PMB = °.
Fill in each blank
Answer:
All the answers
- 1. C → B → A
- 2. B → A → C
- 3. B. OP = OQ, PR = QR, OR = OR, so △OPR ≅ △OQR (SSS)
- 4. 37°
- 5. 15, 9, 90
VIP practice notes and papers
Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.
- Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
- Practice paper: Set out in parts like a school paper, marked, with answers and full working
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