Worked Example: Distance Between Two Points

Find the distance between:

A(2, 3)

B(8, 11)

Step 1: Find the Horizontal Difference

Δx = 8 − 2

Δx = 6

Step 2: Find the Vertical Difference

Δy = 11 − 3

Δy = 8

Step 3: Apply the Pythagorean Theorem

d² = 6² + 8²

d² = 36 + 64

d² = 100

Step 4: Calculate the Distance

d = 10 units

Interpretation

The straight-line distance between the two points is 10 coordinate units.

Worked Example: Negative Coordinates

Find the distance between:

P(−3, 4)

Q(5, −2)

Step 1: Find the Differences

Δx = 5 − (−3) = 8

Δy = −2 − 4 = −6

Step 2: Square the Differences

8² = 64

(−6)² = 36

Step 3: Add

d² = 64 + 36

d² = 100

Step 4: Calculate

d = 10 units

Important Observation

Negative coordinate differences do not produce negative distances because the differences are squared.

Practical Application: Ladders

A ladder leaning against a vertical wall forms a right triangle with the wall and the ground, assuming the wall is vertical and the ground is level.

The ladder is the hypotenuse.

Worked Example

A ladder is 10 m long.

Its base is positioned 6 m from the wall.

How high up the wall does the ladder reach?

Step 1: Identify the Sides

Hypotenuse = 10 m

Horizontal leg = 6 m

Vertical leg = h

Step 2: Apply the Formula

h² + 6² = 10²

Step 3: Calculate

h² + 36 = 100

h² = 64

Step 4: Solve

h = 8 m

Interpretation

The ladder reaches 8 m up the wall.

Reasonableness Check

The vertical height must be shorter than the ladder.

Since 8 m is less than 10 m, the result is reasonable.

Safety Connection

In real situations, ladder placement must also follow appropriate safety guidance. The Pythagorean Theorem determines geometric distances but does not by itself establish whether the ladder is positioned safely.

Practical Application: Ramps

Ramps form right triangles when they connect two different heights across a horizontal distance.

The ramp is the hypotenuse.

Worked Example

A ramp rises 1.2 m vertically over a horizontal distance of 5 m.

Find the ramp length.

Step 1: Identify the Legs

a = 1.2 m

b = 5 m

Step 2: Apply the Formula

c² = 1.2² + 5²

Step 3: Calculate

c² = 1.44 + 25

c² = 26.44

Step 4: Take the Square Root

c ≈ 5.14 m

Interpretation

The ramp must be approximately 5.14 m long along its sloping surface.

Important Observation

The sloping length is greater than the horizontal distance because the ramp also rises vertically.

Practical Application: Television and Screen Sizes

Screen sizes are often described by their diagonal measurement.

The width and height form the legs of a right triangle.

The diagonal is the hypotenuse.

Worked Example

A rectangular screen is:

48 cm wide

36 cm high

Find its diagonal.

Step 1: Apply the Formula

d² = 48² + 36²

Step 2: Calculate

d² = 2304 + 1296

d² = 3600

Step 3: Take the Square Root

d = 60 cm

Interpretation

The screen has a diagonal measurement of 60 cm.

This illustrates why screen dimensions can be described using a single diagonal length.

Practical Application: Construction

Builders use right triangles to check whether corners are square.

A common method uses the 3–4–5 Pythagorean triple.

Worked Example

A builder measures:

3 m along one side of a rectangular foundation.

4 m along the adjacent side.

If the corner is exactly 90°, the diagonal between the endpoints should be:

c² = 3² + 4²

c² = 9 + 16

c² = 25

c = 5 m

Interpretation

If the diagonal measures 5 m, the two measured sides form a right angle, assuming the measurements are accurate.

This technique is useful when setting out rectangular foundations and other structures.

Practical Application: Navigation

The Pythagorean Theorem can calculate straight-line displacement when two movements are perpendicular.

Worked Example

A person walks:

300 m east

400 m north

Find the straight-line distance from the starting point.

Step 1: Identify the Legs

Eastward movement = 300 m

Northward movement = 400 m

Step 2: Apply the Formula

d² = 300² + 400²

Step 3: Calculate

d² = 90000 + 160000

d² = 250000

Step 4: Take the Square Root

d = 500 m

Interpretation

The person finishes 500 m from the starting point.

Distance Versus Displacement

The total distance walked is:

300 + 400 = 700 m

The magnitude of displacement is:

500 m

These values differ because distance measures the length of the path travelled, while displacement describes the straight-line change in position.

Practical Application: Engineering Cables

Support cables often form the hypotenuse of right triangles.

Worked Example

A vertical pole is 12 m tall.

A cable is attached to its top and anchored 5 m from its base.

Find the minimum straight-line cable length.

Step 1: Apply the Formula

c² = 12² + 5²

Step 2: Calculate

c² = 144 + 25

c² = 169

Step 3: Solve

c = 13 m

Interpretation

The straight-line distance from the top of the pole to the anchor point is 13 m.

Additional cable may be required for attachment and tensioning.

Multi-Step Right Triangle Problems

Some geometric problems contain more than one right triangle.

In these cases, we may need to apply the Pythagorean Theorem several times.

Worked Example: Two Connected Right Triangles

A rectangular field measures:

30 m long

40 m wide

A person walks diagonally across the field and then travels 12 m vertically upward along a separate perpendicular route.

Find the straight-line distance from the original starting point to the final point, assuming the 12 m movement is perpendicular to the plane of the field.

Step 1: Find the Field Diagonal

d² = 30² + 40²

d² = 900 + 1600

d² = 2500

d = 50 m

Step 2: Form a Second Right Triangle

The field diagonal is one leg.

The vertical movement is the other leg.

Therefore:

D² = 50² + 12²

Step 3: Calculate

D² = 2500 + 144

D² = 2644

Step 4: Solve

D = √2644 ≈ 51.42 m

Interpretation

The straight-line distance between the starting and final positions is approximately 51.42 m.

Important Observation

The first application of the theorem finds a diagonal in a horizontal plane.

The second application finds a three-dimensional distance.

Right Triangle Geometry in Three Dimensions

The Pythagorean Theorem can also be used to find the space diagonal of a rectangular prism.

For a rectangular prism with:

Length = l

Width = w

Height = h

the space diagonal is:

d = √(l² + w² + h²)

This formula comes from applying the Pythagorean Theorem twice.

Worked Example

A rectangular storage box measures:

Length = 6 cm

Width = 8 cm

Height = 24 cm

Find the distance between opposite corners of the box.

Step 1: Apply the Three-Dimensional Formula

d² = 6² + 8² + 24²

Step 2: Calculate

d² = 36 + 64 + 576

d² = 676

Step 3: Take the Square Root

d = 26 cm

Interpretation

The straight-line distance between opposite corners is 26 cm.

Algebraic Problems Involving Right Triangles

Sometimes side lengths are expressed algebraically.

The Pythagorean Theorem can be used to construct and solve equations.

Worked Example

A right triangle has legs:

x cm

(x + 7) cm

and a hypotenuse of 13 cm.

Find x.

Step 1: Write the Equation

x² + (x + 7)² = 13²

Step 2: Expand

x² + x² + 14x + 49 = 169

Step 3: Simplify

2x² + 14x + 49 = 169

2x² + 14x − 120 = 0

Step 4: Divide by 2

x² + 7x − 60 = 0

Step 5: Factorise

(x + 12)(x − 5) = 0

Therefore:

x = −12 or x = 5

Step 6: Reject the Invalid Solution

A side length cannot be negative.

Therefore:

x = 5 cm

Step 7: Find the Other Leg

x + 7 = 12 cm

Step 8: Verify

5² + 12² = 13²

25 + 144 = 169

169 = 169

The solution is correct.

Combining Right Triangles with Similarity

Right triangles can also be similar.

For example, the triangles with sides:

3, 4, 5

and:

6, 8, 10

are both right triangles.

Their corresponding sides have the same ratio:

6/3 = 2

8/4 = 2

10/5 = 2

Therefore, they are similar by SSS similarity.

This means the second triangle is an enlargement of the first by a scale factor of 2.

Important Connection

The Pythagorean Theorem remains valid when a right triangle is enlarged or reduced proportionally.

If:

a² + b² = c²

then multiplying all side lengths by k gives:

(ka)² + (kb)² = (kc)²

k²a² + k²b² = k²c²

Dividing by k² returns the original relationship.

This shows why all similar right triangles preserve the Pythagorean relationship.

Choosing the Correct Method

When solving a right triangle problem, use the following process.

Identify the Right Angle

Determine whether the triangle contains a 90° angle.

Identify the Hypotenuse

Locate the side opposite the right angle.

Label the Sides

Use a and b for the legs and c for the hypotenuse.

Determine the Unknown

Decide whether you need to find:

• The hypotenuse.

• One of the legs.

• A diagonal distance.

• A coordinate distance.

• Whether the triangle is right-angled.

Select the Formula

For the hypotenuse:

c = √(a² + b²)

For a missing leg:

a = √(c² − b²)

For coordinate distance:

d = √[(x₂ − x₁)² + (y₂ − y₁)²]

Substitute Carefully

Use the correct side lengths and units.

Solve

Square, add or subtract, and take the square root.

Verify

Substitute the calculated value into the original equation.

Interpret

Explain what the result means in the original problem.

Common Mistakes

Using the Pythagorean Theorem on a triangle that is not right-angled

The theorem applies only to right triangles.

Misidentifying the hypotenuse

The hypotenuse is opposite the 90° angle and is always the longest side.

Adding the side lengths instead of their squares

The theorem involves squared lengths.

Forgetting to take the square root

Solving for c² is not the same as solving for c.

Adding instead of subtracting when finding a leg

To find a missing leg, subtract the square of the known leg from the square of the hypotenuse.

Subtracting in the wrong order

The square of the hypotenuse must be the larger quantity.

Accepting negative side lengths

Physical lengths must be positive.

Rounding too early

Keep exact values until the final step whenever possible.

Confusing distance with displacement

The length of a travelled path may be different from the straight-line displacement.

Using inconsistent units

Convert measurements into compatible units before calculating.

Assuming all right triangles have integer side lengths

Many right triangles have irrational side lengths.

Forgetting to check whether the answer is reasonable

The hypotenuse must always be longer than either leg.

Confusing a Pythagorean triple with any three whole numbers

The numbers must satisfy the Pythagorean relationship.

Failing to identify the longest side when testing a triangle

The longest side must be used as c.

Key Terms

• Right Triangle: A triangle containing one angle of exactly 90°.

• Right Angle: An angle measuring 90°.

• Leg: One of the two sides forming the right angle.

• Hypotenuse: The side opposite the right angle and the longest side of a right triangle.

• Pythagorean Theorem: The relationship a² + b² = c² in a right triangle.

• Converse of the Pythagorean Theorem: The principle that a triangle is right-angled if its side lengths satisfy the Pythagorean relationship.

• Pythagorean Triple: Three positive whole numbers satisfying a² + b² = c².

• Primitive Pythagorean Triple: A Pythagorean triple with no common factor greater than 1.

• Square: The result of multiplying a number by itself.

• Square Root: A number that produces a given value when squared.

• Irrational Number: A number that cannot be expressed as a ratio of two integers.

• Exact Value: A value written without rounding.

• Approximate Value: A numerical estimate of an exact quantity.

• Diagonal: A straight line connecting non-adjacent vertices of a polygon or opposite corners of a solid.

• Distance Formula: A formula for calculating the straight-line distance between two coordinate points.

• Coordinate Plane: A two-dimensional system using horizontal and vertical axes.

• Horizontal Distance: Distance measured parallel to the horizontal axis.

• Vertical Distance: Distance measured parallel to the vertical axis.

• Displacement: The straight-line change in position from an initial point to a final point, including direction.

• Distance: The total length of a path or the separation between points, depending on context.

• Acute Triangle: A triangle in which all three angles are less than 90°.

• Obtuse Triangle: A triangle containing one angle greater than 90°.

• Perpendicular Lines: Lines that intersect at a right angle.

• Rectangular Prism: A three-dimensional solid with rectangular faces.

• Space Diagonal: A line connecting opposite vertices of a three-dimensional solid.

• Similar Triangles: Triangles with equal corresponding angles and proportional corresponding sides.

• Scale Factor: The ratio of corresponding lengths in similar figures.

• Verification: Checking a calculated result against the original equation.

• Reasonableness Check: Evaluating whether a mathematical result makes sense.

• Geometric Model: A mathematical representation of a physical shape or situation.

Key Takeaways

• The Pythagorean Theorem describes the relationship between the three sides of a right triangle.

• The two sides forming the right angle are called legs.

• The hypotenuse is opposite the right angle and is always the longest side.

• The Pythagorean Theorem states that a² + b² = c².

• The hypotenuse can be found using c = √(a² + b²).

• A missing leg can be found by subtracting the square of the known leg from the square of the hypotenuse.

• The theorem applies only to right triangles.

• The converse of the theorem can be used to determine whether a triangle is right-angled.

• Pythagorean triples are sets of positive whole numbers satisfying the theorem.

• Common triples include 3–4–5, 5–12–13, and 8–15–17.

• Multiples of Pythagorean triples are also Pythagorean triples.

• Not all right triangles have whole-number side lengths.

• Some side lengths must be expressed using square roots.

• The distance formula is derived from the Pythagorean Theorem.

• Right triangle geometry can be used to calculate distances on coordinate grids.

• The theorem can be applied more than once to solve three-dimensional problems.

• Similar right triangles preserve the Pythagorean relationship.

• Right triangle geometry is useful for ladders, ramps, cables, screens, construction, and navigation.

• Mathematical solutions should include appropriate units and verification.

• The central skill is recognising right triangles, identifying the hypotenuse correctly, and applying the Pythagorean Theorem to calculate unknown distances and side lengths.

Check Your Understanding

Recall and Understanding

1. Define a right triangle and identify its three sides.

2. State the Pythagorean Theorem and explain what each variable represents.

3. Explain why the hypotenuse must always be the longest side.

4. Define a Pythagorean triple and give three examples.

5. Explain how the converse of the Pythagorean Theorem can be used to identify right triangles.

Application

6. A right triangle has legs measuring 9 cm and 12 cm. Calculate the hypotenuse.

7. A right triangle has a hypotenuse of 17 cm and one leg of 8 cm. Find the missing leg.

8. Determine whether a triangle with side lengths 7 cm, 24 cm, and 25 cm is right-angled.

9. Find the distance between points A(1, 2) and B(7, 10).

10. A rectangular field measures 60 m by 80 m. Calculate the diagonal distance across the field.

Analysis and Challenge

11. A 13 m ladder leans against a vertical wall. Its base is 5 m from the wall. How high does the ladder reach?

12. A ramp rises 0.9 m over a horizontal distance of 4.8 m. Calculate its sloping length.

13. A person walks 500 m east and then 1200 m north. Calculate the total distance walked and the magnitude of the displacement.

14. A right triangle has legs measuring x cm and (x + 7) cm, with a hypotenuse of 13 cm. Determine x and verify your solution.

15. A rectangular storage container measures 9 m long, 12 m wide, and 20 m high. Calculate the straight-line distance between opposite corners of the container.

Final Summary

Right triangle geometry provides an important connection between shape, measurement, and algebra.

The Pythagorean Theorem allows unknown side lengths to be calculated using the relationship between the legs and hypotenuse.

It can also be used to determine whether a triangle is right-angled, recognise Pythagorean triples, and calculate distances on coordinate grids.

More advanced applications involve similar triangles, algebraic equations, three-dimensional distances, and practical measurement problems.

The theorem is widely used in architecture, construction, navigation, engineering, and everyday situations involving perpendicular distances.

The central principle is that the square of the hypotenuse of a right triangle equals the sum of the squares of its two legs, providing a reliable method for calculating unknown lengths and solving geometric problems.