7.2 Triangles and Circles
Understanding Circle Theorems
Circle theorems are geometric rules that help us find missing angles and lengths in circles. These are fundamental for solving geometry problems.
Circle theorems are geometric rules that help us find missing angles and lengths in circles. These are fundamental for solving geometry problems.
⚡ The 7 Circle Theorems:
1. Angle in a Semicircle
Always 90°
Triangles drawn from the diameter to the circumference always have a right angle.
2. Angle at Centre
Centre = 2 × Circumference
The angle at the centre is double the angle at the circumference.
3. Same Segment
Angles are Equal
Angles subtended by the same arc are equal, x = x.
4. Cyclic Quadrilateral
Opposite sum to 180°
The opposite angles in a cyclic quadrilateral add up to 180°
5. Tangent & Radius
Meet at 90°
A radius and a tangent line always meet at a right angle.
6. Two Tangents
Equal Length
Two tangents drawn from the same external point are equal in length.
7. Alternate Segment
Angle Between Tangent and Chord = Opposite Angle
The angle between a tangent and a chord equals the angle in the alternate segment.
8. Perpendicular to the chord
Meet at 90°
The perpendicular from the center to a chord, bisects the chord in two halves
Example 1: Angle in a Semicircle
A triangle is inscribed in a semicircle. If one angle is 35°, find the other non-right angle.
Solution:
• Angle at circumference from diameter = 90°
• Sum of angles in triangle = 180°
• $35° + 90° + x = 180°$
• $x = 180° - 125° = 55°$
Answer: The other angle is 55°
• Angle at circumference from diameter = 90°
• Sum of angles in triangle = 180°
• $35° + 90° + x = 180°$
• $x = 180° - 125° = 55°$
Answer: The other angle is 55°
Example 2: Center and Circumference
The angle at the circumference is 42°. Find the angle at the center subtended by the same arc.
Rule: Angle at center = 2 × angle at circumference
Angle at center = $2 \times 42° = 84°$
Answer: The angle at the center is 84°
Rule: Angle at center = 2 × angle at circumference
Angle at center = $2 \times 42° = 84°$
Answer: The angle at the center is 84°
Example 3: Tangent and Radius (Pythagoras)
• Tangent meets radius at 90° → right-angled triangle
• Use Pythagoras: $a^2 + b^2 = c^2$
• $5^2 + 12^2 = c^2$
• $25 + 144 = 169$
• $c = \sqrt{169} = 13$ cm
Answer: The distance is 13 cm
A tangent of length 12 cm touches a circle with radius 5 cm. Find the distance from the center to the end of the tangent.
Solution:• Tangent meets radius at 90° → right-angled triangle
• Use Pythagoras: $a^2 + b^2 = c^2$
• $5^2 + 12^2 = c^2$
• $25 + 144 = 169$
• $c = \sqrt{169} = 13$ cm
Answer: The distance is 13 cm
Example 4: Angles in Same Segment
Two angles are subtended by the same arc. One angle is 48°. Find the other angle.
Rule: Angles in the same segment are equal
Answer: The other angle is also 48°
Rule: Angles in the same segment are equal
Answer: The other angle is also 48°
Example 5: Cyclic Quadrilateral
In a cyclic quadrilateral ABCD, angle A is 75°. Find angle C.
Rule: Opposite angles sum to 180°
Angle C = $180° - 75° = 105°$
Answer: Angle C is 105°
Rule: Opposite angles sum to 180°
Angle C = $180° - 75° = 105°$
Answer: Angle C is 105°
Example 6: Two Tangents from External Point
From point P, two tangents touch a circle. One tangent is 8 cm. Find the other.
Rule: Two tangents from external point are equal
Answer: The other tangent is also 8 cm
Rule: Two tangents from external point are equal
Answer: The other tangent is also 8 cm
Circle Theorems
What is Congruence?
Two shapes are congruent if they are exactly the same shape and size. They might be rotated or reflected, but all corresponding sides and angles are equal.
Two shapes are congruent if they are exactly the same shape and size. They might be rotated or reflected, but all corresponding sides and angles are equal.
⚡ Four Conditions for Triangle Congruence:
SSS
Side-Side-Side
All three sides equal
All three sides equal
SAS
Side-Angle-Side
Two sides and included angle
Two sides and included angle
ASA
Angle-Side-Angle
Two angles and included side
Two angles and included side
RHS
Right-Hypotenuse-Side
Right angle, hypotenuse, one side
Right angle, hypotenuse, one side
Example 1: Identifying SSS Congruence
Triangle ABC has sides 5 cm, 7 cm, and 9 cm.
Triangle DEF has sides 5 cm, 7 cm, and 9 cm.
Solution:
All three corresponding sides are equal.
Conclusion: The triangles are congruent by SSS
Triangle DEF has sides 5 cm, 7 cm, and 9 cm.
Solution:
All three corresponding sides are equal.
Conclusion: The triangles are congruent by SSS
Example 2: Identifying SAS Congruence
Triangle ABC: Sides AB = 6 cm, AC = 8 cm, and angle A = 50°
Triangle XYZ: Sides XY = 6 cm, XZ = 8 cm, and angle X = 50°
Solution:
Two sides are equal (6 cm and 8 cm)
The angle between these sides is equal (50°)
Conclusion: The triangles are congruent by SAS
Triangle XYZ: Sides XY = 6 cm, XZ = 8 cm, and angle X = 50°
Solution:
Two sides are equal (6 cm and 8 cm)
The angle between these sides is equal (50°)
Conclusion: The triangles are congruent by SAS
Example 3: Not Enough Information
Triangle ABC has angles 40°, 60°, and 80°
Triangle DEF has angles 40°, 60°, and 80°
Analysis:
We only know the angles are equal.
We don't know any side lengths.
Conclusion: NOT congruent - these triangles are similar but could be different sizes!
⚠️ Remember: AAA tells us about similarity, not congruence
Triangle DEF has angles 40°, 60°, and 80°
Analysis:
We only know the angles are equal.
We don't know any side lengths.
Conclusion: NOT congruent - these triangles are similar but could be different sizes!
⚠️ Remember: AAA tells us about similarity, not congruence
Congruent Triangles
Same shape, same size
✓ All sides equal
✓ All angles equal
NOT Congruent
Same shape, different size
✗ Different side lengths
(These are similar)
💡 Congruence Checklist:
SSS
All 3 sides matchSAS
2 sides + angle between themASA
2 angles + side between themRHS
Right angle + hypotenuse + 1 side
🎯 Congruence Condition Practice
Understanding Similarity
Two shapes are similar if they have the same angles but are different sizes. One shape is an enlargement or reduction of the other.
Two shapes are similar if they have the same angles but are different sizes. One shape is an enlargement or reduction of the other.
⚡ Scale Factors:
When shapes are similar, we use scale factors to relate their measurements:
Linear Scale Factor (LSF):
$$\text{LSF} = \frac{\text{New Length}}{\text{Original Length}}$$
Area Scale Factor (ASF):
$$\text{ASF} = (\text{LSF})^2$$
Volume Scale Factor (VSF):
$$\text{VSF} = (\text{LSF})^3$$
Linear Scale Factor (LSF):
$$\text{LSF} = \frac{\text{New Length}}{\text{Original Length}}$$
Area Scale Factor (ASF):
$$\text{ASF} = (\text{LSF})^2$$
Volume Scale Factor (VSF):
$$\text{VSF} = (\text{LSF})^3$$
Example 1: Finding Missing Length in Similar Shapes
Two similar triangles. The small triangle has a base of 4 cm and height of 6 cm. The large triangle has a base of 12 cm. Find its height.
→
Step 1: Find the scale factor
$\text{LSF} = \frac{12}{4} = 3$
Step 2: Apply to height
New height $= 6 \times 3 = 18$ cm
Answer: The height of the large triangle is 18 cm
Small Triangle
Large Triangle
$\text{LSF} = \frac{12}{4} = 3$
Step 2: Apply to height
New height $= 6 \times 3 = 18$ cm
Answer: The height of the large triangle is 18 cm
Example 2: Area Scale Factor
Two similar rectangles. The smaller has dimensions 3 cm × 4 cm. The larger has dimensions 9 cm × 12 cm. Compare their areas.
Step 1: Find LSF
$\text{LSF} = \frac{9}{3} = 3$
Step 2: Calculate areas
Small area $= 3 \times 4 = 12$ cm²
Large area $= 9 \times 12 = 108$ cm²
Step 3: Find ASF
$\text{ASF} = (\text{LSF})^2 = 3^2 = 9$
Check: $108 \div 12 = 9$ ✓
Answer: The large rectangle has 9 times the area
Step 1: Find LSF
$\text{LSF} = \frac{9}{3} = 3$
Step 2: Calculate areas
Small area $= 3 \times 4 = 12$ cm²
Large area $= 9 \times 12 = 108$ cm²
Step 3: Find ASF
$\text{ASF} = (\text{LSF})^2 = 3^2 = 9$
Check: $108 \div 12 = 9$ ✓
Answer: The large rectangle has 9 times the area
Example 3: Volume Scale Factor
Two similar cylinders. The small cylinder has radius 2 cm and volume 60 cm³. The large cylinder has radius 6 cm. Find its volume.
Step 1: Find LSF
$\text{LSF} = \frac{6}{2} = 3$
Step 2: Find VSF
$\text{VSF} = (\text{LSF})^3 = 3^3 = 27$
Step 3: Calculate large volume
Large volume $= 60 \times 27 = 1620$ cm³
Answer: The large cylinder has volume 1620 cm³
Step 1: Find LSF
$\text{LSF} = \frac{6}{2} = 3$
Step 2: Find VSF
$\text{VSF} = (\text{LSF})^3 = 3^3 = 27$
Step 3: Calculate large volume
Large volume $= 60 \times 27 = 1620$ cm³
Answer: The large cylinder has volume 1620 cm³
💡 Key Differences:
| Property | Congruent | Similar |
|---|---|---|
| Shape | Same ✓ | Same ✓ |
| Size | Same ✓ | Different ✗ |
| Angles | Same ✓ | Same ✓ |
| Side Lengths | Same ✓ | Proportional |
| Scale Factor | 1 | Any value |
🎯 Similar Shapes Practice
Real Life Uses:
• Architecture: Scale models of buildings (e.g., 1:100 scale)
• Maps: Representing large areas on paper with scale factors
• Photography: Zoom and crop operations create similar images
• Manufacturing: Creating different sizes of the same product
• Cooking: Scaling recipes and cake tin sizes
• Art: Enlarging or reducing drawings and designs
• Architecture: Scale models of buildings (e.g., 1:100 scale)
• Maps: Representing large areas on paper with scale factors
• Photography: Zoom and crop operations create similar images
• Manufacturing: Creating different sizes of the same product
• Cooking: Scaling recipes and cake tin sizes
• Art: Enlarging or reducing drawings and designs