IB MYP 3 Mathematics 6.5 Geometric Problem Solving Study Notes - New Syllabus
IB MYP 3 Mathematics 6.5 Geometric Problem Solving Study Notes
IB MYP 3 Mathematics 6.5 Geometric Problem Solving Study Notes at IITian Academy focus on specific topics and types of questions asked in the actual exam. Study Notes focus on the IB MYP 3 Mathematics syllabus with guiding questions of
- Parallel lines → use corresponding, alternate and co-interior angle relationships.
- Triangles → use angle sum, isosceles properties, equilateral properties and right-angle properties.
- Quadrilaterals and polygons → use angle sums and special-shape properties.
- Congruent triangles → use equal corresponding sides and angles.
- Similar triangles → use equal corresponding angles and proportional corresponding sides.
- Right-angled triangles → use Pythagoras’ theorem.
6.5 – Geometric Problem Solving
Geometric problem solving brings together the ideas from Topics 6.1–6.4. Instead of using one theorem in isolation, many geometry problems require you to combine angle facts, triangle properties, polygon rules, congruence, similarity, and Pythagoras’ theorem.
What Is Geometric Problem Solving?
A geometric problem often gives you a diagram containing several pieces of information. Some information may be given directly, while other information must be calculated.
For example, a diagram might contain:
- parallel lines,
- right angles,
- equal sides,
- equal angles,
- triangles,
- similar or congruent shapes,
- known lengths,
- unknown angles or lengths.
Your task is to determine which facts are connected and use them in the correct order.
The Geometry Problem-Solving Cycle
- Read the question carefully.
- Inspect the diagram for markings and relationships.
- Identify what is known and what is unknown.
- Choose the theorem or property that connects them.
- Calculate the new information.
- Use that information in the next step if necessary.
- Check whether the answer is reasonable.
- State the final answer clearly with units where appropriate.
Read the Diagram Before Calculating
A geometry diagram contains important clues. Before doing any calculations, look for the following markings.
| Diagram Marking | Meaning |
|---|---|
| Small square at an angle | The angle is \(90^\circ\) |
| Matching tick marks on sides | The sides are equal in length |
| Matching arcs on angles | The angles are equal |
| Arrow markings on lines | The lines are parallel |
| Same labelled length | Those lengths are equal |
⚠️ Important
Do not assume that two lines are parallel, two sides are equal, or two angles are equal simply because the diagram looks that way. Use the information and markings provided.
Angle Facts to Use
Many geometric problems begin with finding one or more missing angles. These basic angle facts should become automatic.
| Relationship | Rule |
|---|---|
| Angles on a straight line | Sum is \(180^\circ\) |
| Angles around a point | Sum is \(360^\circ\) |
| Vertically opposite angles | They are equal |
| Angles in a triangle | Sum is \(180^\circ\) |
| Angles in a quadrilateral | Sum is \(360^\circ\) |
Parallel Lines and Transversals
When a transversal crosses two parallel lines, several pairs of angles have predictable relationships.
| Angle Pair | Relationship |
|---|---|
| Corresponding angles | Equal |
| Alternate interior angles | Equal |
| Co-interior angles | Sum to \(180^\circ\) |
Example 1:
Two parallel lines are crossed by a transversal. One of the angles formed is \(65^\circ\). A triangle is formed between the lines, and another angle of the triangle is \(50^\circ\). Find the remaining angle of the triangle.
▶️ Answer/Explanation
Use the parallel-line relationship first to determine the required angle of the triangle. Suppose this gives \(65^\circ\).
The angles in a triangle sum to \(180^\circ\):
\(x=180^\circ-115^\circ\)
\(x=65^\circ\)
Therefore, the missing angle is \(65^\circ\).
Triangle Problem Solving
Triangles are often the basic building blocks of larger geometric problems. Look for opportunities to use their angle and side properties.
Important Triangle Properties
- The angles in a triangle sum to \(180^\circ\).
- An isosceles triangle has two equal sides and two equal base angles.
- An equilateral triangle has three equal sides and three \(60^\circ\) angles.
- A right-angled triangle has one \(90^\circ\) angle.
- The hypotenuse of a right-angled triangle is opposite the \(90^\circ\) angle.
Example 2:
An isosceles triangle has a vertex angle of \(40^\circ\). Find each base angle.
▶️ Answer/Explanation
The two base angles are equal. Let each one be \(x\).
\(2x=140^\circ\)
\(x=70^\circ\)
Each base angle is \(70^\circ\).
Quadrilaterals and Polygons
Larger polygons can often be solved by breaking them into triangles or using known angle-sum formulas.
📌 Interior Angle Sum
For a polygon with \(n\) sides:
For a regular polygon, all interior angles are equal. Therefore:
Useful Quadrilateral Properties
| Quadrilateral | Important Properties |
|---|---|
| Parallelogram | Opposite sides are parallel and equal; opposite angles are equal. |
| Rectangle | Four right angles; opposite sides equal. |
| Rhombus | Four equal sides; opposite angles equal. |
| Square | Four equal sides and four right angles. |
| Trapezium | One pair of parallel sides. |
Congruent Triangles in Problem Solving
Two triangles are congruent if they have exactly the same shape and size.
Congruence allows us to transfer known lengths and angles from one triangle to another.
Congruence Tests
- SSS: three corresponding sides are equal.
- SAS: two corresponding sides and the included angle are equal.
- ASA: two corresponding angles and the included side are equal.
- RHS: right angle, hypotenuse and one corresponding side are equal.
Once two triangles have been shown to be congruent, their corresponding sides and angles are equal.
Example 3:
Two triangles are congruent. A side of the first triangle is \(8\) cm. The corresponding side of the second triangle is \(x+2\) cm. Find \(x\).
▶️ Answer/Explanation
Corresponding sides of congruent triangles are equal.
\(x=6\)
Therefore, \(x=6\).
Similar Triangles
Similar triangles have the same shape, but they do not necessarily have the same size.
Corresponding angles are equal, while corresponding sides are in the same ratio.
Scale Factor
The scale factor from one similar shape to another is:
Example 4:
Two triangles are similar. A side of the smaller triangle is \(6\) cm and the corresponding side of the larger triangle is \(15\) cm. Another side of the smaller triangle is \(8\) cm. Find the corresponding side of the larger triangle.
▶️ Answer/Explanation
First find the scale factor:
Multiply the corresponding side by \(2.5\):
Therefore, the corresponding side is \(20\) cm.
Similarity and Indirect Measurement
Similar triangles are especially useful when a length cannot be measured directly.
For example, the height of a tree or building can be found by comparing it with a smaller object that forms a similar triangle.
Example 5:
A \(1.5\) m tall object casts a \(2\) m shadow. At the same time, a tree casts a \(10\) m shadow. Assuming the triangles are similar, find the height of the tree.
▶️ Answer/Explanation
Similar triangles have proportional corresponding sides.
Therefore:
\(x=7.5\)
Therefore, the tree is \(7.5\) m tall.
Pythagoras in Geometric Problems
Pythagoras’ theorem can be used whenever a right-angled triangle is present:
In a complicated diagram, the right-angled triangle may not be obvious. You may need to draw an auxiliary line or split the shape into simpler triangles.
Look for:
- Rectangles and squares containing diagonals.
- Perpendicular sides.
- Vertical and horizontal distances.
- Right-angled triangles hidden inside larger shapes.
- Lengths that can become a side of another triangle.
Example 6:
A rectangle is \(9\) cm long and \(12\) cm wide. Find the length of its diagonal and then find the perimeter of the rectangle.
▶️ Answer/Explanation
The diagonal forms the hypotenuse of a right-angled triangle.
\(d^2=81+144\)
\(d^2=225\)
\(d=15\)
Now calculate the perimeter:
\(P=42\text{ cm}\)
The diagonal is \(15\) cm and the perimeter is \(42\) cm.
Combining Similarity and Pythagoras
Some of the more challenging problems require more than one theorem. A common combination is similarity followed by Pythagoras.
🧠 Strategy
- Use similarity to determine an unknown side.
- Use the newly calculated side in a right-angled triangle.
- Apply Pythagoras’ theorem.
- Check the final answer against the diagram.
Working Backwards
Some geometry problems give you the final information and ask you to determine an earlier unknown.
In these situations, work backwards from the quantity you need.
🔙 Working-Backwards Method
- Ask: What do I need to find?
- Ask: Which theorem can find it?
- Ask: What information does that theorem require?
- Find that missing information first.
- Return to the original quantity.
Introducing Variables into Geometry
Geometry problems often use algebraic expressions for unknown lengths or angles.
For example, suppose two angles in a triangle are \(2x+10^\circ\) and \(x+20^\circ\), while the third angle is \(60^\circ\).
Example 7:
The angles of a triangle are \(2x+10^\circ\), \(x+20^\circ\), and \(60^\circ\). Find \(x\).
▶️ Answer/Explanation
The angles in a triangle sum to \(180^\circ\):
\(3x+90=180\)
\(3x=90\)
\(x=30\)
Therefore, \(x=30\).
Choosing the Correct Theorem
When you are unsure what to do, first identify the type of information in the problem.
| What You See | Think About |
|---|---|
| Two parallel lines | Corresponding, alternate or co-interior angles |
| Triangle | \(180^\circ\), isosceles properties, equilateral properties |
| Quadrilateral | \(360^\circ\) and special quadrilateral properties |
| Polygon | \((n-2)\times180^\circ\) |
| Matching side/angle markings | Congruence |
| Same shape, different sizes | Similarity and scale factor |
| Right-angled triangle | Pythagoras’ theorem |
| Three sides given, right angle unknown | Converse of Pythagoras’ theorem |
Multi-Step Geometry
The hardest problems in this topic may require several different ideas. Do not try to solve the entire problem in one step.
🪜 A Reliable Multi-Step Strategy
- Mark every known angle and length.
- Identify all right angles.
- Look for parallel lines.
- Find any easy angles first.
- Look for congruent or similar triangles.
- Calculate missing lengths using similarity or Pythagoras.
- Use the newly found information in the next step.
- Continue until the required answer is obtained.
Example 8:
A rectangular garden has length \(12\) m and width \(5\) m. A diagonal path is constructed from one corner to the opposite corner. Find the length of the path and the angle the path makes with the \(12\) m side, correct to 1 decimal place.
▶️ Answer/Explanation
First, the diagonal forms a right-angled triangle.
Let the diagonal be \(d\).
\(d^2=144+25\)
\(d^2=169\)
\(d=13 \)
Therefore, the diagonal path is \(13\) m long.
The triangle is a \(5\)-\(12\)-\(13\) right triangle. To find the angle, we would use the appropriate trigonometric relationship once trigonometry is part of the course.
At this level, the key geometric step is recognizing that the rectangle’s diagonal creates a right-angled triangle and applying Pythagoras correctly.
Checking Your Answer
Geometry problems should always include a quick reasonableness check.
- Is an angle between \(0^\circ\) and \(180^\circ\) for a triangle?
- Do the angles of a triangle add to \(180^\circ\)?
- Do the angles of a quadrilateral add to \(360^\circ\)?
- Is the hypotenuse longer than either shorter side?
- Does the scale factor make sense?
- Does the answer have the correct unit?
- Does the answer look reasonable compared with the diagram?
🔎 Example of a Quick Check
If the two shorter sides of a right-angled triangle are \(6\) cm and \(8\) cm, the hypotenuse must be greater than \(8\) cm.
Since:
\(10\) cm makes sense because it is longer than both \(6\) cm and \(8\) cm.
⚠️ Common Errors to Avoid
- Assuming the diagram is drawn to scale.
- Using a theorem without checking that its conditions are satisfied.
- Choosing the wrong corresponding sides in similar triangles.
- Mixing up congruent and similar triangles.
- Forgetting that the hypotenuse is opposite the \(90^\circ\) angle.
- Using Pythagoras on a triangle that is not right angled.
- Forgetting to use parallel-line angle relationships.
- Forgetting that angles in a triangle total \(180^\circ\).
- Using the polygon angle formula incorrectly.
- Rounding too early in a multi-step problem.
- Leaving out units for lengths.
- Giving an answer without showing the reasoning.
Example 9:
A rectangular field is \(24\) m long and \(10\) m wide. A diagonal path connects two opposite corners.
a) Find the length of the diagonal.
b) Explain why Pythagoras’ theorem can be used.
c) The diagonal divides the rectangle into two triangles. Explain why these two triangles are congruent.
▶️ Answer/Explanation
a) The diagonal creates a right-angled triangle because adjacent sides of a rectangle meet at \(90^\circ\).
Let the diagonal be \(d\).
\(d^2=576+100\)
\(d^2=676\)
\(d=26\)
The diagonal is \(26\) m.
b) Pythagoras’ theorem applies because the diagonal and the two sides form a right-angled triangle.
c) The two triangles have equal corresponding sides: the two \(24\) m sides are equal, the two \(10\) m sides are equal, and they share the same diagonal. Therefore, the triangles are congruent by SSS.