Class 10 bpt theorem

BPT Theorem ( Basic Proportionality Theorem) or Thale’s Theorem – Theorem no 6.1 Theorem 6.1 is : If a line is drawn parallel to one side of a triangle intersecting the other two sides. Then it divides the two sides in the same ratio. BPT Theorem or Thale’s Theorem detailed step-by-step explanation is given in the below image and YouTube video. Figure 1: BPT or Thale’s Theorem

Transcript AA Criteria If two angles of one triangle are respectively equal to two angles of another triangle, then the two triangles are similar.Given: Two triangles ∆ABC and ∆DEF such that ∠B = ∠E & ∠C = ∠F To Prove: ∆ABC ~ ∆DEF Proof: In ∆ ABC, By angle sum property ∠A + ∠B + ∠C = 180° In ∆ DEF, By angle sum property ∠D + ∠E + ∠F = 180° In ∆ DEF, By angle sum property ∠D + ∠E + ∠F = 180° From (1) and (2) ∠A + ∠B + ∠C = ∠D + ∠E + ∠F ∠A + ∠E + ∠F = ∠D + ∠E + ∠F ∠ A = ∠ D Thus, In Δ ABC & Δ DEF ∠ A = ∠ D ∠ B = ∠ E ∠ C = ∠ F ∴ Δ ABC ~ Δ DEF Hence, proved Show More

Transcript Theorem 6.1: If a line is drawn parallel to one side of a triangle to intersect the other two side in distinct points, the other two sides are divided in the same ratio. Given: Δ ABC where DE ∥ BC To Prove: 𝐴𝐷/𝐷𝐵 = 𝐴𝐸/𝐸𝐶 Construction: Join BE and CD Draw DM ⊥ AC and EN ⊥ AB. Proof: Now, Now, ar (ADE) = 1/2 × Base × Height = 1/2 × AE × DM ar (DEC) = 1/2 × Base × Height = 1/2 × EC × DM Divide (3) and (4) "ar (ADE)" /"ar (DEC)" = (1/2 " × AE × DM" )/(1/2 " × EC × DM " ) "ar (ADE)" /"ar (DEC)" = "AE" /"EC" Now, ∆BDE and ∆DEC are on the same base DE and between the same parallel lines BC and DE. ∴ ar (BDE) = ar (DEC) Hence, "ar (ADE)" /"ar (BDE)" = "ar (ADE)" /"ar (DEC)""AD" /"DB" = "AE" /"EC" Hence Proved. Show More

• ICSE Solutions • ICSE Solutions for Class 10 • ICSE Solutions for Class 9 • ICSE Solutions for Class 8 • ICSE Solutions for Class 7 • ICSE Solutions for Class 6 • Selina Solutions • ML Aggarwal Solutions • ISC & ICSE Papers • ICSE Previous Year Question Papers Class 10 • ISC Previous Year Question Papers • ICSE Specimen Paper 2021-2022 Class 10 Solved • ICSE Specimen Papers 2020 for Class 9 • ISC Specimen Papers 2020 for Class 12 • ISC Specimen Papers 2020 for Class 11 • ICSE Time Table 2020 Class 10 • ISC Time Table 2020 Class 12 • Maths • Merit Batch Math Labs with Activity – Proportionality Theorem, or Thales Theorem OBJECTIVE To verify the basic proportionality theorem, or Thales theorem Materials Required • A sheet of whitepaper • A sheet of colored paper • A piece of cardboard • A tube of glue • A ruler • A geometry box Theory Basic proportionality theorem (or Thales theorem): If a line is drawn parallel to one side of a triangle intersecting the other two sides then the line divides these sides in the same ratio. Procedure Step 1: Paste the sheet of white paper on the cardboard. Step 2: Draw a straight line PQ = 10 cm on this paper. At P draw a line MP perpendicular to PQ and at Q draw a line NQ perpendicular to PQ. Graduate the two lines MP and NQ as shown in Figure 6.1. Step 3: Draw a line XY parallel to PQ by taking points X and Y at an equal distance from P and Q respectively, and then joining them. Step 4: Cut a triangle ABC from the colored paper and paste i...

Triangle is a plane figure consisting of three sides, and three angles opposite to the sides makes a sum of 180 degrees. It comes in the category of a polygon; polygon is a closed shape plane with line segments. Triangle is the simplest polygon in geometry. Triangles theorems have significant use in geometry; they prove various properties associated with it. To give students in-depth knowledge about triangle theorems, Vedantu has uploaded the well-researched explanations on the website. Understanding the theorems is necessary to solve the problems linked with triangles. Properties of a Triangle The properties of a triangle include the followings: • It has three sides, angles, and vertices • The sum of three interior angles are always 180 degree • The sum of the two sides of this geometrical figure is greater than its third one • The area of the product of this figure’s height and the base is equal to twice its area. Types of Triangle There are different According to the Measurement of Angles • Acute angle, where all interior angles are less than 90 degrees. • Right angle, where one of the three interior angles of a triangle is 90 degrees. • Obtuse angle, where one interior angle is greater than 90 degrees. According to the Measurement of Sides • An equilateral triangle is where all 3 sides are equal. • An isosceles triangle is where 2 sides are equal. • A scalene triangle is where no sides are equal. Since the definition of triangles and their types are now clear, students...

Triangle is a plane figure consisting of three sides, and three angles opposite to the sides makes a sum of 180 degrees. It comes in the category of a polygon; polygon is a closed shape plane with line segments. Triangle is the simplest polygon in geometry. Triangles theorems have significant use in geometry; they prove various properties associated with it. To give students in-depth knowledge about triangle theorems, Vedantu has uploaded the well-researched explanations on the website. Understanding the theorems is necessary to solve the problems linked with triangles. Properties of a Triangle The properties of a triangle include the followings: • It has three sides, angles, and vertices • The sum of three interior angles are always 180 degree • The sum of the two sides of this geometrical figure is greater than its third one • The area of the product of this figure’s height and the base is equal to twice its area. Types of Triangle There are different According to the Measurement of Angles • Acute angle, where all interior angles are less than 90 degrees. • Right angle, where one of the three interior angles of a triangle is 90 degrees. • Obtuse angle, where one interior angle is greater than 90 degrees. According to the Measurement of Sides • An equilateral triangle is where all 3 sides are equal. • An isosceles triangle is where 2 sides are equal. • A scalene triangle is where no sides are equal. Since the definition of triangles and their types are now clear, students...

Basic Proportionality Theorem & Similar Triangles Basic Proportionality theorem was introduced by a famous Greek Mathematician, Thales, hence it is also called Thales Theorem. According to him, for any two equiangular triangles, the ratio of any two corresponding sides is always the same. Based on this concept, he gave theorem of basic proportionality (BPT). This concept has been introduced in • i) Corresponding angles of both the triangles are equal • ii) Corresponding sides of both the triangles are in proportion to each other Learn Table of contents: • • • • • • Thus two triangles ΔABC and ΔPQR are similar if, • i) ∠A=∠P, ∠B=∠Q and ∠C=∠R • ii) AB/PQ, BC/QR, AC/PR Also, read: • Property of Triangles • Parallel Lines • Important Questions For Class 6 Maths Thales Theorem Statement Let us now state the Basic Proportionality Theorem which is as follows: If a line is drawn parallel to one side of a triangle intersecting the other two sides in distinct points, then the other two sides are divided in the same ratio. Basic Proportionality Theorem Proof Let us now try to prove the basic proportionality theorem statement Consider a triangle ΔABC, as shown in the given figure. In this triangle, we draw a line PQ parallel to the side BC of ΔABC and intersecting the sides AB and AC in P and Q, respectively. According to the basic proportionality theorem as stated above, we need to prove: AP/PB = AQ/QC Construction Join the vertex B of ΔABC to Q and the vertex C to...

BPT Theorem ( Basic Proportionality Theorem) or Thale’s Theorem – Theorem no 6.1 Theorem 6.1 is : If a line is drawn parallel to one side of a triangle intersecting the other two sides. Then it divides the two sides in the same ratio. BPT Theorem or Thale’s Theorem detailed step-by-step explanation is given in the below image and YouTube video. Figure 1: BPT or Thale’s Theorem

Transcript AA Criteria If two angles of one triangle are respectively equal to two angles of another triangle, then the two triangles are similar.Given: Two triangles ∆ABC and ∆DEF such that ∠B = ∠E & ∠C = ∠F To Prove: ∆ABC ~ ∆DEF Proof: In ∆ ABC, By angle sum property ∠A + ∠B + ∠C = 180° In ∆ DEF, By angle sum property ∠D + ∠E + ∠F = 180° In ∆ DEF, By angle sum property ∠D + ∠E + ∠F = 180° From (1) and (2) ∠A + ∠B + ∠C = ∠D + ∠E + ∠F ∠A + ∠E + ∠F = ∠D + ∠E + ∠F ∠ A = ∠ D Thus, In Δ ABC & Δ DEF ∠ A = ∠ D ∠ B = ∠ E ∠ C = ∠ F ∴ Δ ABC ~ Δ DEF Hence, proved Show More

Transcript Theorem 6.1: If a line is drawn parallel to one side of a triangle to intersect the other two side in distinct points, the other two sides are divided in the same ratio. Given: Δ ABC where DE ∥ BC To Prove: 𝐴𝐷/𝐷𝐵 = 𝐴𝐸/𝐸𝐶 Construction: Join BE and CD Draw DM ⊥ AC and EN ⊥ AB. Proof: Now, Now, ar (ADE) = 1/2 × Base × Height = 1/2 × AE × DM ar (DEC) = 1/2 × Base × Height = 1/2 × EC × DM Divide (3) and (4) "ar (ADE)" /"ar (DEC)" = (1/2 " × AE × DM" )/(1/2 " × EC × DM " ) "ar (ADE)" /"ar (DEC)" = "AE" /"EC" Now, ∆BDE and ∆DEC are on the same base DE and between the same parallel lines BC and DE. ∴ ar (BDE) = ar (DEC) Hence, "ar (ADE)" /"ar (BDE)" = "ar (ADE)" /"ar (DEC)""AD" /"DB" = "AE" /"EC" Hence Proved. Show More