Class 10 Mathematics Chapter 6 Triangles

Key Concept: Theorem Application

b) 7.5 cm
[Solution Description]
Using Theorem 6.1, which states $\frac{PS}{SQ} = \frac{PT}{TR}$, we substitute the known values:$\frac{3}{5} = \frac{4.5}{TR}$

Cross-multiplying gives:$3 \cdot TR = 4.5 \cdot 5$

Simplifying,$3 \cdot TR = 22.5$

Dividing both sides by 3,$TR = \frac{22.5}{3} = 7.5 \text{ cm}$
Hence, the length of TR is 7.5 cm.

Notes

Key Concept: Real-World Application

a) It is parallel to the riverbank.
[Solution Description] By the converse of the Basic Proportionality Theorem, since line XY divides the segments in equal ratio, XY must be parallel to AB.

Notes

• Finance: Mathematics is used in calculating interest rates for savings, loans, and investments. Simple and compound interest formulas help in financial planning.
• Architecture and Engineering: Geometry and trigonometry help in designing buildings, bridges, and structures, ensuring stability and precision.
• Navigation: Mathematics is used for calculating distances, directions, and positions, crucial in map-making, GPS, and navigation systems.
• Medicine: Statistics and probability help in analyzing medical data, conducting research, and predicting disease outbreaks.
• Population Studies: Growth rates, models, and projections are based on mathematical concepts to understand demographic changes.
• Technology: Algorithms, coding, and data analysis in computer science heavily rely on mathematical principles.

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Key Concept: Proportional Sides

c) 12 cm and 18 cm
[Solution Description] Since the rectangles are similar, their corresponding sides are in proportion. Let the sides of the second rectangle be 4x cm and 6x cm. The scale factor must remain constant: $\frac{4}{4x} = \frac{6}{6x}$, which gives us x = 1. Therefore, possible dimensions based on different values of x can include any multiple that maintains the same ratio 4:6, such as 8 cm and 12 cm or 12 cm and 18 cm.

Notes

• Definition of Proportionality: If two triangles are similar, their corresponding sides are in the same ratio (proportional).
• Similarity Criterion: Triangles are similar if their corresponding angles are equal and sides are proportional (AAA, SAS, or SSS criteria).
• Basic Proportionality Theorem (Thales’ Theorem): A line drawn parallel to one side of a triangle divides the other two sides proportionally.
• Conversely: If a line divides two sides of a triangle in the same ratio, it is parallel to the third side.
• Applications: Used in solving problems involving triangles, geometry, and real-life measurements.

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Key Concept: Proportional Segments

c) 8 cm
[Solution Description] The side lengths of similar triangles are proportional. Given AB:DE = 3:4 and AB = 6 cm, then:

$\frac{6}{DE} = \frac{3}{4}$

Cross-multiplying gives:$3 \cdot DE = 6 \cdot 4$

$3 \cdot DE = 24$

Dividing both sides by 3:$DE = \frac{24}{3} = 8$

Therefore, the length of DE is 8 cm.

Notes

• Basic Proportionality Theorem (Thales’ Theorem): If a line is drawn parallel to one side of a triangle, it divides the other two sides in the same ratio. $\frac{AB}{BC} = \frac{DE}{EF}$
• Converse of Basic Proportionality Theorem: If a line divides two sides of a triangle in the same ratio, it is parallel to the third side.
• Applications: Used in solving problems involving similar triangles and finding unknown side lengths.
• Key Properties: Proportionality holds true only if the parallel line cuts both sides.
• Practical Tip: Check parallelism before applying the theorem.

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Key Concept: Converse Reasoning

a) The line is parallel to the third side
[Solution Description] By applying the converse of the Basic Proportionality Theorem, if a line divides two sides of a triangle in the same ratio, it implies that the line is parallel to the third side.

Notes

• Converse Theorem: If a condition of a theorem is proven true in reverse, it’s called its converse.
• Converse of Pythagoras Theorem: If the square of one side equals the sum of squares of the other two sides, the triangle is a right triangle.
• Converse of Basic Proportionality Theorem (BPT): If a line divides two sides proportionally, it is parallel to the third side.
• Logical Validation: Converse reasoning ensures accuracy by reversing original theorems.
• Applications: Used in solving geometric problems and proofs.

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Key Concept: Correspondence Identification

c) P’Q’
[Solution Description]
In similar figures, corresponding sides are those that occupy the same relative positions. Thus, side PQ in pentagon PQRST corresponds to side P’Q’ in pentagon P’Q’R’S’T’.

Notes

Key Concept: Converse Application

c) EF is parallel to QR
[Solution Description]
By the converse of the Basic Proportionality Theorem, if a line divides any two sides of a triangle in the same ratio, then it is parallel to the third side.
Therefore, since $\frac{PE}{EQ} = \frac{PF}{FR}$, line EF is parallel to side QR.

Notes

• Definition: Converse of a theorem reverses the conditions and conclusion. In triangles, converse applications are used to validate properties of shapes.
• Key Converse Theorems:
  • Pythagoras Theorem: If
    $a^2 + b^2 = c^2$, then the triangle is right-angled.
  • Basic Proportionality Theorem (BPT): If a line divides two sides proportionally, it is parallel to the third side.
• Application:
  • Verifying shapes as right-angled or similar triangles.
  • Proving parallelism or proportionality.
• Importance: Converse helps in reverse reasoning for solving proofs and problems.

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Key Concept: Proportionality Application

b) 25 cm
[Solution Description] The distance on the map can be calculated using the scale. Given the scale is 1 cm for every 10 km, the distance on the map will be:
$\frac{250 \text{ km}}{10 \text{ km/cm}} = 25 \text{ cm}$

Notes

• Basic Proportionality Theorem (BPT): If a line is parallel to one side of a triangle and intersects the other two sides, it divides those sides proportionally.
• Converse of BPT: If a line divides two sides of a triangle proportionally, then it is parallel to the third side.
• Application: Proportionality theorem helps in solving problems involving areas, lengths, and angles of similar triangles.
• Important Theorems:
  • Area ratio of two similar triangles is the square of the ratio of their corresponding sides.
  • The altitude to the hypotenuse of a right triangle divides it into two smaller triangles that are similar to each other and the original triangle.

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Key Concept: Angle Correspondence

b) $\angle E$
[Solution Description] In similar triangles, corresponding angles are equal. Since $\Delta ABC \sim \Delta DEF$, the second angle of both triangles corresponds to each other. Thus, $\angle B$ corresponds to $\angle E$.

Notes

• Angle Correspondence: Refers to the relationship between the angles when two parallel lines are intersected by a transversal.
• Types of Corresponding Angles:
  • Alternate Angles: Angles on opposite sides of the transversal but inside the parallel lines.
  • Corresponding Angles: Angles in the same position relative to the parallel lines and transversal.
  • Co-interior Angles: Angles on the same side of the transversal and inside the parallel lines.
  • Exterior Angles: Angles outside the parallel lines, formed by the transversal.
• Angle Properties:
  • Alternate interior angles are equal.
  • Corresponding angles are equal.
  • Co-interior angles are supplementary (sum to 180°).

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Key Concept: SSS Ratio Calculation

d) Triangles are similar
[Solution Description] To check for SSS similarity, we find the ratios of corresponding sides. For triangles ABC and DEF, calculate:

$\frac{AB}{DE} = \frac{6}{9} = \frac{2}{3}, \quad\frac{BC}{EF} = \frac{8}{12} = \frac{2}{3}, \quad\frac{CA}{FD} = \frac{10}{15} = \frac{2}{3}$

Since all three pairs of sides have the same ratio $\frac{2}{3}$, the triangles are similar by the SSS criterion.

Notes

• SSS Criterion: The SSS (Side-Side-Side) similarity criterion states that if the corresponding sides of two triangles are proportional, then the triangles are similar.
• • If triangles
    △ABC $\triangle$ ABCand △DEF $\triangle$ DEF

    are similar, then:Formula for SSS Ratio:

  ABDE=BCEF=CAFD​

• Steps for Calculation:
  • Identify corresponding sides of the two triangles.
  • Measure or use the given side lengths.
  • Find the ratio between the corresponding sides.
• Use in Problems: SSS ratio helps in proving the similarity of triangles and solving for unknown side lengths.

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Key Concept: Proportional Sides

a) Yes, they are similar.
[Solution Description] To determine if the triangles are similar using the SSS criterion, we must check if their corresponding sides are proportional. Computing the ratios:

$\frac{7}{14} = \frac{24}{48} = \frac{25}{50} = \frac{1}{2}$

Since all corresponding side ratios are equal, triangles ABC and DEF are similar by the SSS similarity criterion.

Notes

• Basic Definition: Two triangles are said to have proportional sides if the corresponding sides of one triangle are proportional to the corresponding sides of the other triangle.​
• AA Criterion: If two triangles have two corresponding angles equal, then their corresponding sides are proportional.
• Side-Angle-Side (SAS) Similarity: If one angle of a triangle is equal to the corresponding angle of another triangle, and the sides including these angles are proportional, the triangles are similar.
• Basic Proportionality Theorem: If a line is parallel to one side of a triangle and intersects the other two sides, it divides those sides proportionally.
• Application: Proportional sides help in finding unknown lengths in similar triangles.
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Key Concept: Polygon Comparison

a) Yes
[Solution Description] For two rectangles to be similar, their corresponding angles must be equal (which they already are as all angles in a rectangle are 90 degrees) and the ratios of their corresponding sides must be equal. Here, the sides are in the correct ratio 5:7, satisfying the similarity condition. Therefore, the rectangles are similar.

Notes

• Definition: Polygons are closed figures made of straight lines. Comparison involves analyzing sides, angles, and properties.
• Similar Polygons: Two polygons are similar if their corresponding angles are equal and sides are proportional.
• Criteria for Similarity:
  • AA (Angle-Angle): Two corresponding angles are equal.
  • SSS (Side-Side-Side): All corresponding sides are proportional.
  • SAS (Side-Angle-Side): Two sides are proportional, and the included angle is equal.
• Applications: Helps in solving real-world problems like indirect measurements.
• Key Formula: Ratio of areas of two similar polygons = (Ratio of corresponding sides)².

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Key Concept: Basic Proportionality Theorem Application

b) 4/3
[Solution Description] By the Basic Proportionality Theorem, since OP is parallel to MN, $\frac{LO}{OM} = \frac{LP}{PN}$. Thus, $\frac{LP}{PN} = \frac{4}{3}$.

Notes

• Statement of BPT: If a line is drawn parallel to one side of a triangle and intersects the other two sides, it divides those sides in the same ratio.
  • AB/BC = PQ/QR (for ΔABC with line PQ ∥ BC).
• Key Applications:
  • Finding Unknown Lengths: Solve proportional equations to determine missing side lengths.
  • Prove Similarity: Use BPT to prove triangles similar by AA criterion.
  • Geometric Proofs: Establish relationships between line segments in divided triangles.
• Real-life Uses: Applied in maps, models, and architectural designs.

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Key Concept: Fill in the Blanks

b) Similar
[Solution Description] Equilateral triangles have all sides equal and all angles equal to $60^\circ$, hence they are similar by definition.

Notes

• A triangle is said to be similar if their corresponding angles are equal and corresponding sides are in proportion.
• The Basic Proportionality Theorem (Thales Theorem) states that if a line is drawn parallel to one side of a triangle, it divides the other two sides in the same ratio.
• In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides, known as Pythagoras Theorem.
• Two triangles are similar if their corresponding sides are in proportion, called SAS Similarity Criterion.
• AA Criterion requires two corresponding angles to be equal for triangle similarity.

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Key Concept: Real-World Application, Indirect Measurement

c) 12.5 meters
[Solution Description] To solve this problem, we need to use indirect measurement through similar triangles. The ratios of the corresponding sides of these triangles should be equal because they share the same angle caused by the sun. Thus, the proportion would be:

$\frac{\text{Height of flagpole}}{\text{Length of flagpole’s shadow}} = \frac{\text{Height of building}}{\text{Length of building’s shadow}}$

Let’s denote the height of the flagpole as  h . So our equation becomes:$\frac{h}{5} = \frac{50}{20}$

By cross-multiplying, we get:$h \times 20 = 50 \times 5$

Simplifying gives us:20h = 250

Dividing both sides by 20, we find:$h = \frac{250}{20} = 12.5$

Therefore, the height of the flagpole is 12.5 meters.

Notes

Key Concept: Symbolic Representation

b) $\sim$
[Solution Description] The symbol used to denote similarity between two geometric figures is “~”. This symbol indicates that the shapes are the same but may vary in size.

Notes

• Symbolic Representation involves using symbols, such as letters, to represent numbers or quantities.
• Algebraic Expressions: A combination of numbers, variables (letters), and mathematical operations like addition, subtraction, multiplication, and division.
• Variables: Symbols that stand for unknown values, often represented by letters like x, y, or z.
• Coefficients: Numbers multiplying variables in an algebraic expression.
• Like Terms: Terms that have the same variables raised to the same powers (e.g., 3x and 5x).
• Constant Terms: Numbers without variables.
• Simplification: Combining like terms to reduce expressions to their simplest form.

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Key Concept: Converse Proof

b) Theorem 6.2
[Solution Description] Given $PS/SQ = PT/TR$ and $\angle PSQ = \angle PTR$, we need to determine which theorem supports the conclusion that STR is an isosceles triangle.

By Theorem 6.2, if a line divides two sides of a triangle in the same ratio, then the line is parallel to the third side. This establishes that ST || RQ.

Since ST is parallel to RQ and equal angles are subtended (given), triangle STR is thus isosceles.

Notes

• Statement: In a triangle, if the square of one side equals the sum of the squares of the other two sides, the triangle is a right triangle.
• Given: A triangle ABC where $ AB^2 + BC^2 = AC^2 $
• To Prove: ∠B=90∘
• Construction: Draw
  △DEF where ∠E=90∘ and sides DE=AB , EF=BC

• Proof: Use Pythagoras Theorem in
  △DEF and show DF=AC
• Conclusion:
  △ABC≅△DEF by SSS, hence ∠B=90∘

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Key Concept: Converse Application

(a) Their angles must be equal

[Solution Description]
When two triangles have proportional sides according to the SSS criterion, it implies that the triangles are similar. This means their corresponding angles are also equal.

Notes

• Converse of Similarity Theorems: If the sides of one triangle are proportional to the sides of another triangle, then the triangles are similar.
• SSS Criterion: If all three pairs of corresponding sides are in the same ratio, the triangles are similar.
• SAS Criterion: If one angle is equal and the sides including the angle are proportional, the triangles are similar.
• Application: Used in solving geometric problems involving indirect measurements, constructing parallel lines, and verifying similarity.
• Importance: Fundamental for proofs and constructions in advanced geometry.

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Key Concept: Missing Side

b) 5 cm
[Solution Description] Let the missing leg be b. Using the Pythagorean Theorem:
$12^2 + b^2 = 13^2$
Calculate the squares:$144 + b^2 = 169$
Solving for $b^2$ gives:$b^2 = 169 – 144 = 25$
Therefore:$b = 5\, \text{cm}$

Notes                           

1. Pythagoras Theorem: In a right-angled triangle, the square of the hypotenuse (side opposite the right angle) is equal to the sum of the squares of the other two sides. Formula: $c^2 = a^2 + b^2$where c is the hypotenuse, and a, b are the other sides.
2. Using Similar Triangles: For two similar triangles, corresponding sides are proportional.
   $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$
3. Basic Proportionality Theorem: If a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally.

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Key Concept: Theorem Application

c) 8 cm
[Solution Description]
According to Theorem 6.1, since DE is parallel to BC, the sides are divided in the same ratio, so:
$\frac{AD}{DB} = \frac{AE}{EC}$
Substituting the known values gives:$\frac{3}{6} = \frac{4}{EC}$
Simplifying $\frac{3}{6}$ gives $\frac{1}{2}$. So,
$\frac{1}{2} = \frac{4}{EC}$
Cross-multiply to solve for EC:$1 \cdot EC = 2 \cdot 4$
EC = 8  cm

Notes

• Introduction to Theorem Application: Theorems in geometry are used to solve practical problems. These include applications of congruence, similarity, and properties of circles.
• Basic Theorems: Key theorems like Pythagoras Theorem, Properties of Triangles, and Circle Theorems are foundational.
• Pythagoras Theorem: States that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
• Congruence & Similarity: Theorems related to congruent triangles (SSS, SAS, ASA, RHS) and similar triangles (AA, SAS, SSS) help in real-life applications.
• Circle Theorems: Tangent properties, angles subtended by chords, and cyclic quadrilaterals are crucial in solving geometric problems.
• Application: These theorems are applied in construction, design, navigation, and real-world measurements.

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Key Concept: Missing Side in Triangle

c) 10 feet
[Solution Description]
Using Pythagoras Theorem: $ c^2 = a^2 + b^2 $

Let the ladder be the hypotenuse  c = 26  ft, the height up the wall  a = 24 ft, and  b  the distance from the base to the wall.

Substitute the values:$26^2 = 24^2 + b^2$

Calculate the squares:$676 = 576 + b^2$

Subtract 576 from both sides:$b^2 = 100$

Find the square root:$b = \sqrt{100} = 10$

Hence, the base of the ladder is 10 feet from the wall.

Notes

• Pythagoras Theorem: In a right-angled triangle, $a^2 + b^2 = c^2$, where cc is the hypotenuse. Use this to find the missing side.
• Types of Triangles:
  • Right-Angled: Use Pythagoras Theorem.
  • Other triangles: Use rules like the triangle inequality.
• Triangle Inequality Rule: The sum of any two sides must be greater than the third side.
• Trigonometry: Use sine, cosine, or tangent ratios when angles are known.
• Similarity of Triangles: Proportions of corresponding sides help determine the missing side.
• Practice Example: Apply these rules to solve triangles systematically.

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Key Concept: Segment Calculation

b) 8 cm
[Solution Description] By the Basic Proportionality Theorem, we know that if a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally.

Using the theorem: $\frac{AD}{DB} = \frac{AE}{EC}$

Substitute the known values: $\frac{3}{4} = \frac{6}{EC}$

Cross-multiply to solve for EC: $3 \cdot EC = 4 \cdot 6$ $3 \cdot EC = 24$

Divide both sides by 3:EC = $\frac{24}{3}$ = 8 cm

Notes

• Segment Definition: A segment is a part of a line connecting two points. In triangles, segments like medians, altitudes, and sides are calculated using geometric principles.
• Midpoint Formula: Used to find the midpoint of a segment:(x₁​+x₂,y₁+y_2​)
• Distance Formula: To calculate segment length:d= (x₂​−x₁​)²+(y₂​−y₁​)²
  
• Basic Proportionality Theorem: If a line parallel to one side of a triangle intersects the other two sides, it divides them proportionally.
• Applications: Used in coordinate geometry, proofs, and similarity concepts.

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Key Concept: RHS Criterion

a) Yes, they are similar.
[Solution Description]
The triangles are right-angled, and for the RHS criterion, we check if the hypotenuses and one other side are proportional. Here, SU = VX and the side proportion is ST/VW = 1/2. Thus, according to the RHS criterion, the triangles $\triangle STU$ and $\triangle VWX$ are similar because the hypotenuses and a pair of corresponding sides are proportional.

Notes

• Definition: RHS (Right Angle-Hypotenuse-Side) criterion is a rule to prove the congruence of two right triangles.
• Conditions:
  • Both triangles must have a right angle.
  • Their hypotenuses (longest sides) must be equal.
  • One pair of corresponding sides (other than the hypotenuse) must be equal.
• Congruence Statement: If the above conditions are satisfied, the two triangles are congruent (ΔABC ≅ ΔDEF).
• Reasoning: Based on the unique properties of right triangles and congruence rules.
• Applications: Used in solving problems involving construction, proofs, and geometrical relationships.

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Key Concept: Prove Parallelism

a) Line $DE \parallel BC$
[Solution Description] The section formula states that if a line divides two sides of a triangle proportionally, then it is parallel to the third side. Given that $AD/DB = AE/EC$, by converse of Basic Proportionality Theorem, we can conclude that line $DE \parallel BC$.

Notes

Key Concept: Real-World Application

b) 30 m
[Solution Description]
Let the height of the building be $h$. By using the concept of similar triangles formed by the shadows, we have:
$\frac{10}{15} = \frac{h}{45}$
Solving for h:$h = \frac{10 \times 45}{15} = 30 \text{ m}$
Therefore, the height of the building is 30 meters.

Notes

• Construction and Architecture: Triangles are used in building bridges, towers, and roofs for stability and weight distribution.
• Navigation and Mapping: Triangulation helps determine distances and locations using angles and distances between points.
• Art and Design: Triangular patterns are widely used in graphic design and art for aesthetic appeal.
• Surveying Land: Triangles help calculate inaccessible distances, like the height of a hill or the width of a river.
• Engineering: Structural designs use triangular frameworks to ensure rigidity and strength in vehicles and machinery.
• Astronomy: Triangles assist in measuring distances between celestial objects.

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Key Concept: AAA Criterion

c) They are similar.
[Solution Description] According to the AAA criterion, if all corresponding angles of two triangles are equal, their corresponding sides will be in the same ratio, making them similar.

Notes

• Definition: The AAA (Angle-Angle-Angle) Criterion states that if three angles of one triangle are equal to the three corresponding angles of another triangle, the two triangles are similar.
• Similarity: Triangles are similar if:
  • Their corresponding angles are equal.
  • Their corresponding sides are in proportion.
• Conditions:
  • Only angles need to match; side lengths can vary but maintain the same ratio.
• Importance: Used to prove similarity in geometric problems without measuring sides.
• Example: If ∠A = ∠P, ∠B = ∠Q, and ∠C = ∠R, then △ABC ~ △PQR.

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