Solved Exercises on Triangles and Triangle Properties in Grade 8

Master triangles and triangle properties: classification, angle sum, Pythagorean theorem, area, and similarity through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Triangle Classification by Sides
Exercise 1
A triangle has sides of length 5 cm, 5 cm, and 8 cm. Classify the triangle by its sides and find its perimeter.
Definition:

Triangle classification by sides: Equilateral (3 equal sides), Isosceles (2 equal sides), Scalene (no equal sides)

Solution method:
  1. Compare the lengths of all three sides
  2. Determine which classification applies
  3. Calculate the perimeter by adding all side lengths
Given
Sides: 5, 5, 8 cm
Classification
Isosceles
Perimeter
18 cm
Step 1: Compare side lengths

Side 1 = 5 cm, Side 2 = 5 cm, Side 3 = 8 cm

Two sides are equal (5 cm each)

Step 2: Apply classification rule

Since exactly two sides are equal, the triangle is isosceles

Step 3: Calculate perimeter

Perimeter = Side 1 + Side 2 + Side 3

Perimeter = 5 + 5 + 8 = 18 cm

Isosceles triangle with perimeter 18 cm
Final answer:

The triangle is isosceles with a perimeter of 18 cm

Applied rules:

• Triangle classification: Equilateral (3 equal), Isosceles (2 equal), Scalene (none equal)

• Perimeter: Sum of all side lengths

• Verification: Check that 5 + 5 + 8 = 18 ✓

2 Triangle Angle Sum Theorem
Exercise 2
In a triangle, two angles measure 40° and 70°. Find the measure of the third angle and classify the triangle by its angles.
Definition:

Triangle angle sum theorem: The sum of the interior angles of any triangle is always 180°

Given
40° + 70° + x = 180°
Simplify
110° + x = 180°
Solution
x = 70°
Step 1: Apply angle sum theorem

Sum of all three angles = 180°

40° + 70° + Third angle = 180°

Step 2: Solve for the third angle

Third angle = 180° - 40° - 70° = 70°

Step 3: Classify by angles

All angles are less than 90°, so it's an acute triangle

Two angles are equal (70° each), so it's also isosceles

Third angle = 70°, Acute Isosceles triangle
Final answer:

The third angle measures 70°, making it an acute isosceles triangle

Applied rules:

• Angle sum theorem: Interior angles sum to 180°

• Triangle classification: Acute (all < 90°), Right (one = 90°), Obtuse (one > 90°)

• Verification: Check that 40° + 70° + 70° = 180° ✓

3 Pythagorean Theorem
Exercise 3
A right triangle has legs measuring 6 cm and 8 cm. Find the length of the hypotenuse.
Definition:

Pythagorean theorem: In a right triangle, a² + b² = c², where c is the hypotenuse

Given
a = 6, b = 8
Formula
c² = a² + b²
Solution
c = 10 cm
Step 1: Identify the parts of the right triangle

Legs: a = 6 cm, b = 8 cm

Hypotenuse: c = ? cm (unknown)

Step 2: Apply Pythagorean theorem

c² = a² + b²

c² = 6² + 8²

Step 3: Calculate the squares

c² = 36 + 64 = 100

Step 4: Find the hypotenuse

c = √100 = 10 cm

Hypotenuse = 10 cm
Final answer:

The hypotenuse measures 10 cm

Applied rules:

• Pythagorean theorem: a² + b² = c² for right triangles

• Algebra: Solve for unknown variable

• Verification: Check that 6² + 8² = 10² (36 + 64 = 100) ✓

Rules and methods, laws,...
a + b + c = 180°
Triangle Angle Sum
a² + b² = c²
Pythagorean Theorem
A = ½bh
Triangle Area
By Sides
Equilateral, Isosceles, Scalene
Based on side lengths
By Angles
Acute, Right, Obtuse
Based on angle measures
Triangle Inequality
a + b > c
Sum of any two sides > third side
Key definitions:

Triangle: A polygon with three sides and three angles

Equilateral triangle: All three sides equal, all angles 60°

Isosceles triangle: Two sides equal, base angles equal

Scalene triangle: No sides equal

Acute triangle: All angles less than 90°

Right triangle: One angle equals 90°

Obtuse triangle: One angle greater than 90°

Hypotenuse: Longest side of a right triangle (opposite the right angle)

Legs: The two shorter sides of a right triangle

Complete methodology:
  1. Identify the triangle type: By sides or angles
  2. Select the appropriate theorem: Angle sum, Pythagorean, area, etc.
  3. Set up the equation: Based on the theorem
  4. Solve for the unknown: Using algebraic methods
  5. Verify the solution: Check against the original conditions
Tip 1: Remember the angle sum is always 180° in any triangle.
Tip 2: In a right triangle, the hypotenuse is always the longest side.
Tip 3: An equilateral triangle is also equiangular (all 60°).
Tip 4: The Pythagorean theorem only works for right triangles.
Common errors: Forgetting the angle sum is 180°, misapplying the Pythagorean theorem to non-right triangles, confusing legs with hypotenuse.
Exam preparation: Memorize triangle classifications, practice the Pythagorean theorem, learn area formulas, understand triangle inequality.
Formulas to know by heart:

• Triangle angle sum: ∠A + ∠B + ∠C = 180°

• Pythagorean theorem: a² + b² = c²

• Triangle area: A = ½bh

• Triangle inequality: Sum of any two sides > third side

• Isosceles triangle: Base angles are equal

• Equilateral triangle: All sides equal, all angles 60°

Solution: Exercises 4 to 5
4 Triangle Area
Exercise 4
A triangle has a base of 12 cm and a height of 8 cm. Find its area. Then, if the base is doubled while the height remains the same, how does the area change?
Definition:

Triangle area: A = ½ × base × height, where height is perpendicular to the base

Original
A = ½(12)(8)
Calculate
A = 48 cm²
New base
24 cm
Step 1: Apply area formula (original triangle)

A = ½ × base × height

A = ½ × 12 × 8 = 48 cm²

Step 2: Calculate new area (doubled base)

New base = 2 × 12 = 24 cm

New area = ½ × 24 × 8 = 96 cm²

Step 3: Compare areas

Original area = 48 cm²

New area = 96 cm²

The area doubled when the base was doubled

Original area = 48 cm², New area = 96 cm²
Final answer:

The original area is 48 cm². When the base is doubled, the area doubles to 96 cm².

Applied rules:

• Area formula: A = ½bh

• Proportional reasoning: Area is directly proportional to the base

• Verification: Check calculations and compare results

5 Triangle Similarity
Exercise 5
Triangle ABC has sides of 3 cm, 4 cm, and 5 cm. Triangle DEF has sides of 6 cm, 8 cm, and 10 cm. Are the triangles similar? If so, what is the scale factor?
Definition:

Similar triangles: Triangles with equal corresponding angles and proportional corresponding sides

Triangle ABC
3, 4, 5 cm
Triangle DEF
6, 8, 10 cm
Scale factor
2:1
Step 1: List the corresponding sides

ABC: 3, 4, 5 cm

DEF: 6, 8, 10 cm

Step 2: Check if sides are proportional

6/3 = 2, 8/4 = 2, 10/5 = 2

All ratios are equal to 2

Step 3: Verify it's a right triangle (optional check)

ABC: 3² + 4² = 9 + 16 = 25 = 5² ✓

DEF: 6² + 8² = 36 + 64 = 100 = 10² ✓

Step 4: State conclusion

Since all corresponding sides are proportional with the same ratio, the triangles are similar

Scale factor = 2:1 (DEF is twice the size of ABC)

Triangles are similar with scale factor 2:1
Final answer:

Yes, the triangles are similar with a scale factor of 2:1.

Applied rules:

• Similarity criterion: All corresponding sides are proportional

• Scale factor: Ratio of corresponding sides

• Verification: Check that ratios are consistent

Key Concepts, Laws, Methods, and Formulas for Triangles
a + b + c = 180°
Triangle Angle Sum
Key definitions:

Triangle: A polygon with three sides, three vertices, and three interior angles

Vertex: The point where two sides of a triangle meet

Side: A line segment connecting two vertices of a triangle

Interior angle: An angle inside the triangle formed by two sides

Exterior angle: An angle formed by extending one side of the triangle

Altitude: A perpendicular line segment from a vertex to the opposite side

Median: A line segment from a vertex to the midpoint of the opposite side

Angle bisector: A line that divides an angle into two equal parts

Equilateral triangle: All three sides equal in length, all angles equal to 60°

Isosceles triangle: At least two sides equal in length, base angles equal

Scalene triangle: All three sides of different lengths

Acute triangle: All three angles less than 90°

Right triangle: One angle exactly 90°, the longest side is the hypotenuse

Obtuse triangle: One angle greater than 90°

Complete methodology:
  1. Classify the triangle: Determine if it's by sides (equilateral, isosceles, scalene) or by angles (acute, right, obtuse)
  2. Identify what's given: Which measurements or relationships are provided
  3. Choose the appropriate theorem: Angle sum, Pythagorean, area, similarity, etc.
  4. Set up the equation: Based on the chosen theorem and given information
  5. Solve systematically: Using algebraic methods to find unknown values
  6. Verify the solution: Check that the answer makes sense in the context
Tip 1: Remember that the sum of any two sides must be greater than the third side (Triangle Inequality).
Tip 2: In an isosceles triangle, the base angles (angles opposite the equal sides) are always equal.
Tip 3: The altitude to the base of an isosceles triangle bisects both the base and the vertex angle.
Tip 4: In a right triangle, the Pythagorean theorem (a² + b² = c²) always holds.
Tip 5: The area of a triangle is half the area of a rectangle with the same base and height.
Tip 6: When triangles are similar, their corresponding angles are equal and their sides are proportional.
Common errors: Forgetting the angle sum is 180°, misapplying the Pythagorean theorem to non-right triangles, confusing the hypotenuse with the legs in right triangles, misidentifying corresponding parts in similar triangles.
Memory aids: SOHCAHTOA for trigonometric ratios, "All Students Take Calculus" for quadrant signs, "Please Excuse My Dear Aunt Sally" for order of operations in calculations.
Problem-solving strategies: Draw accurate diagrams, label all known information, use geometric properties to find unknowns, check that your answer makes sense in the context of the diagram.
Essential formulas and theorems:

• Triangle angle sum: ∠A + ∠B + ∠C = 180°

• Exterior angle theorem: An exterior angle equals the sum of the two remote interior angles

• Pythagorean theorem: a² + b² = c² (for right triangles only)

• Triangle area: A = ½bh

• Heron's formula: A = √[s(s-a)(s-b)(s-c)] where s = semi-perimeter

• Triangle inequality: a + b > c, a + c > b, b + c > a

• Isosceles triangle theorem: Base angles are equal

• Converse of isosceles theorem: If two angles are equal, the opposite sides are equal

• Similar triangles: Corresponding angles equal, corresponding sides proportional

• Congruent triangles: SSS, SAS, ASA, AAS, HL criteria

Visual Representation: Triangle Properties
Exercise 6: Triangle Classification Comparison
Visual representation of different triangle classifications:
- By sides: Equilateral, Isosceles, Scalene
- By angles: Acute, Right, Obtuse
- Area relationships
- Special properties

Analysis: The chart illustrates how different triangle properties relate to each other and their classifications.

  • Equilateral triangles are always acute (all angles 60°)
  • Right triangles follow the Pythagorean theorem
  • Isosceles triangles have equal base angles
  • Area depends on base and height regardless of type

Questions & Answers

Question: I'm confused about when to use the Pythagorean theorem versus other triangle formulas. How do I know which one to use?

Answer: Great question! Here's how to decide:

  • Pythagorean theorem (a² + b² = c²): Only use this for RIGHT triangles when you know two sides and need the third
  • Triangle angle sum (180°): Use when you know some angles and need others
  • Area formula (A = ½bh): Use when you need to find the area given base and height
  • Triangle inequality: Use to determine if three lengths can form a triangle

Before applying any formula, ask yourself: "What kind of triangle is this?" and "What am I trying to find?" The Pythagorean theorem is very specific - it only works for right triangles.

Example: If you have a triangle with sides 3, 4, 5, you can verify it's a right triangle using Pythagorean theorem: 3² + 4² = 9 + 16 = 25 = 5².

Question: Can a triangle be both isosceles and right? Or isosceles and equilateral?

Answer: Yes to the first question, no to the second:

  • Isosceles AND right: Yes! This is a right triangle with two equal sides (legs). The angles would be 45°, 45°, and 90°.
  • Isosceles AND equilateral: Technically yes, but an equilateral triangle is a special case of isosceles. However, the standard definition of isosceles is "at least two equal sides," so equilateral fits this definition.

More specifically:

  • An isosceles right triangle has sides in the ratio 1:1:√2
  • An equilateral triangle is also isosceles, but an isosceles triangle is not necessarily equilateral
  • An equilateral triangle cannot be right since all angles are 60°

Remember: An equilateral triangle is a special case of an isosceles triangle, but not all isosceles triangles are equilateral.

Question: What's the difference between similar and congruent triangles? They seem like the same thing.

Answer: Great question! They're related but different:

  • Congruent triangles: Same shape AND same size. All corresponding sides and angles are equal.
  • Similar triangles: Same shape but NOT necessarily the same size. Corresponding angles are equal, and sides are proportional.

Think of it this way:

  • Congruent = Identical copies (exact same dimensions)
  • Similar = Enlarged or reduced copies (same shape, different size)

Criteria:

  • Congruent: SSS, SAS, ASA, AAS, HL
  • Similar: AA, SSS (proportional), SAS (proportional)

If two triangles are congruent, they are automatically similar (with scale factor 1), but similar triangles are not necessarily congruent.

Question: How do I know if three given lengths can form a triangle?

Answer: You use the Triangle Inequality Theorem:

Triangle Inequality: The sum of the lengths of any two sides must be greater than the length of the third side.

For three lengths a, b, and c to form a triangle, all three conditions must be met:

  • a + b > c
  • a + c > b
  • b + c > a

Example: Can lengths 3, 4, and 8 form a triangle?

  • 3 + 4 = 7, and 7 is not greater than 8
  • Since one condition fails, these lengths cannot form a triangle

If any of these inequalities is not satisfied, the three lengths cannot form a triangle.

Question: Why is the area of a triangle half the area of a rectangle with the same base and height?

Answer: This is a beautiful geometric relationship! Here's why:

  • If you draw a diagonal across a rectangle, it divides the rectangle into two identical triangles
  • Each triangle has the same base and height as the rectangle
  • Since the rectangle is split into two equal parts, each triangle has half the area of the rectangle

Mathematically:

  • Rectangle area = base × height
  • Triangle area = ½ × base × height

This works for any triangle because you can always imagine it as half of a parallelogram (which is essentially a "tilted" rectangle). The height must always be measured perpendicular to the base.

This is why the formula is A = ½bh regardless of the triangle's shape - the relationship to the rectangle remains constant.