Converse of bpt theorem class 10th

Given, in \(\triangle ABC\), D and E are the mid points of AB and AC respectively, such that, AD=BD and AE=EC. We have to prove that: DE || BC. Since, D is the midpoint of AB AD=DB =>\(\frac\) By converse of Basic Proportionality Theorem, DE || BC Hence, proved. NCERT solutions of related questions for Triangles • In figure. (i) and (ii), DE || BC. Find EC in (i) and AD in (ii). • E and F are points on the sides PQ and PR respectively of a \(\triangle PQR\). For each of the following cases, state whether EF || QR. (i) PE = 3.9 cm, EQ = 3 cm, PF = 3.6 cm and FR = 2.4 cm (ii) PE = 4 cm, QE = 4.5 cm, PF = 8 cm and RF = 9 cm (iii) PQ = 1.28 cm, PR = 2.56 cm, PE = 0.18 cm and PF = 0.63 cm • In the figure, if LM || CB and LN || CD, prove that \(\frac\). Show that ABCD is a trapezium.

Basic Proportionality Theorem Basic proportionality theorem was proposed by a famous Greek mathematician,Thales, hence, it is also referred to as the Thales theorem.According to the famous mathematician, for any two equiangular triangles, the ratio of any two corresponding sides of the given triangles is always the same. Based on this concept, the basic proportionality theorem(BPT) was proposed. It gives the relationship between the sides ofany two equiangular triangles. Theconcept of Thales theoremhas been introduced insimilar triangles.If the giventwo triangles are similar to each other then, • Corresponding angles of both the triangles are equal • Corresponding sides of both the triangles are in proportion to each other The theorem thus also helps us better understand the concept of similar triangles.Now let us try and understand the Basic Proportionality Theorem. 1. 2. 3. 4. Proof of the Basic Proportionality Theorem Let us nowtry to prove the basic proportionality(BPT) theoremstatement. Statement:The line drawnparallelto one side of a Given:Consider a triangle ΔABC, as shown in the given figure. In this triangle, we draw a line DEparallel to the side BC of ΔABC and intersecting the sides AB and AC atDand E, respectively. Construction:In the above diagram, create imaginary lines where you can Join C to D and B to E. Draw Proof: Considerthe triangles ADE and BDE. Both these triangles are onthe same base AB and have equal height EQ. (Area of ADE)/(Area of BDE)= (1/2× AD ...

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...

More • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • Maths Theorems There are several maths theorems which govern the rules of modern mathematics. Almost in every branch of mathematics, there are numerous theorems established by renowned mathematicians from around the world. Here, the list of most important theorems in maths for all the classes (from 6 to 12) are provided, which are essential to build a stronger foundation in basic mathematics. To consider a mathematical statement as a theorem, it requires proof. The proof confirms that the given mathematical statement is true. Proving the theorems develops logical thinking and reasoning skills along with avoiding Maths errors. List of Maths Theorems What are Theorems in Maths? Mathematical theorems can be defined as statements which are accepted as true through prev...

In the past, a famous Greek mathematician named Thales came up with a fact relating to two equiangular triangles. The crucial truth provided by Thales states that the ratio of any two corresponding sides of two equiangular triangles is always the same. Based on this concept, the mathematician later gave a theorem known as Basic Proportionality Theorem (BPT) or Thales theorem. Here, we will state and prove BPT, which is the key to understanding the above-stated concept introduced in similar triangles better. Thales Theorem Statement Thales or Basic Proportionality Theorem States: If a line is drawn parallel to one side of a triangle to intersect its other two sides in distinct points, then the other two sides are divided in the same ratio. Applying the statement on the triangle ABC (given below), we can say that if the line segment DE is drawn parallel to the side BC of triangle ABC to intersect sides AB and AC respectively, then DE divides AB and AC in the same ratio. Proof of the Basic Proportionality Theorem Given, 1. Triangle ABC 2. DE ∥ BC To Prove: According to the BPT stated above, we need to prove: AD/DB = AE/EC Construction: From vertex B, draw a line meeting the side AC of triangle ABC at E to form a line BE. Now, from E draw a perpendicular EN to the side AB. Similarly, join the vertex C to D to form a line CD and then, draw a perpendicular DM to the side AC, as shown in the figure. Proof: Consider triangle ADE and recall the formula for the area of a triangle, w...

Transcript 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. Given: Δ ABC and a line DE intersecting AB at D and AC at E, such that "AD" /"DB" = "AE" /"EC" To Prove: DE ∥ BC Construction: Draw DE’ parallel to BC. Proof: Since DE’ ∥ BC , By Theorem 6.1 :If a line is drawn parallel to one side of a triangle to intersecting other two sides not distinct points, the other two sided are divided in the same ratio. ∴ 𝐴𝐷/𝐷𝐵 = (𝐴𝐸^′)/(𝐸^′ 𝐶) And given that, 𝐴𝐷/𝐷𝐵 = 𝐴𝐸/𝐸𝐶 From (1) and (2) (𝐴𝐸^′)/(𝐸^′ 𝐶) = 𝐴𝐸/𝐸𝐶 Adding 1 on both sides (𝐴𝐸^′)/(𝐸^′ 𝐶) + 1 = 𝐴𝐸/𝐸𝐶 + 1 (𝐴𝐸^′ + 𝐸^′ 𝐶)/(𝐸^′ 𝐶) = (𝐴𝐸 + 𝐸𝐶)/𝐸𝐶 "AE" /"EC" + 1 = "AE′" /"E′C" + 1 ("AE" + "EC" )/"EC" = ("AE′" + "E′C" )/"E′C""AC" /"EC" = "AC" /"E′C" EC = E’C Thus, E and E’ coincides. Hence, DE ∥ BC. (𝐴𝐸^′ + 𝐸^′ 𝐶)/(𝐸^′ 𝐶) = (𝐴𝐸 + 𝐸𝐶)/𝐸𝐶 𝐴𝐶/(𝐸^′ 𝐶) = 𝐴𝐶/𝐸𝐶 1/(𝐸^′ 𝐶) = 1/𝐸𝐶 EC = E’C Thus, E and E’ coincide Since DE’ ∥ BC ∴ DE ∥ BC. Hence, proved Show More

Basic Proportionality Theorem Basic proportionality theorem was proposed by a famous Greek mathematician,Thales, hence, it is also referred to as the Thales theorem.According to the famous mathematician, for any two equiangular triangles, the ratio of any two corresponding sides of the given triangles is always the same. Based on this concept, the basic proportionality theorem(BPT) was proposed. It gives the relationship between the sides ofany two equiangular triangles. Theconcept of Thales theoremhas been introduced insimilar triangles.If the giventwo triangles are similar to each other then, • Corresponding angles of both the triangles are equal • Corresponding sides of both the triangles are in proportion to each other The theorem thus also helps us better understand the concept of similar triangles.Now let us try and understand the Basic Proportionality Theorem. 1. 2. 3. 4. Proof of the Basic Proportionality Theorem Let us nowtry to prove the basic proportionality(BPT) theoremstatement. Statement:The line drawnparallelto one side of a Given:Consider a triangle ΔABC, as shown in the given figure. In this triangle, we draw a line DEparallel to the side BC of ΔABC and intersecting the sides AB and AC atDand E, respectively. Construction:In the above diagram, create imaginary lines where you can Join C to D and B to E. Draw Proof: Considerthe triangles ADE and BDE. Both these triangles are onthe same base AB and have equal height EQ. (Area of ADE)/(Area of BDE)= (1/2× AD ...

In the past, a famous Greek mathematician named Thales came up with a fact relating to two equiangular triangles. The crucial truth provided by Thales states that the ratio of any two corresponding sides of two equiangular triangles is always the same. Based on this concept, the mathematician later gave a theorem known as Basic Proportionality Theorem (BPT) or Thales theorem. Here, we will state and prove BPT, which is the key to understanding the above-stated concept introduced in similar triangles better. Thales Theorem Statement Thales or Basic Proportionality Theorem States: If a line is drawn parallel to one side of a triangle to intersect its other two sides in distinct points, then the other two sides are divided in the same ratio. Applying the statement on the triangle ABC (given below), we can say that if the line segment DE is drawn parallel to the side BC of triangle ABC to intersect sides AB and AC respectively, then DE divides AB and AC in the same ratio. Proof of the Basic Proportionality Theorem Given, 1. Triangle ABC 2. DE ∥ BC To Prove: According to the BPT stated above, we need to prove: AD/DB = AE/EC Construction: From vertex B, draw a line meeting the side AC of triangle ABC at E to form a line BE. Now, from E draw a perpendicular EN to the side AB. Similarly, join the vertex C to D to form a line CD and then, draw a perpendicular DM to the side AC, as shown in the figure. Proof: Consider triangle ADE and recall the formula for the area of a triangle, w...

More • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • Maths Theorems There are several maths theorems which govern the rules of modern mathematics. Almost in every branch of mathematics, there are numerous theorems established by renowned mathematicians from around the world. Here, the list of most important theorems in maths for all the classes (from 6 to 12) are provided, which are essential to build a stronger foundation in basic mathematics. To consider a mathematical statement as a theorem, it requires proof. The proof confirms that the given mathematical statement is true. Proving the theorems develops logical thinking and reasoning skills along with avoiding Maths errors. List of Maths Theorems What are Theorems in Maths? Mathematical theorems can be defined as statements which are accepted as true through prev...

Given, in \(\triangle ABC\), D and E are the mid points of AB and AC respectively, such that, AD=BD and AE=EC. We have to prove that: DE || BC. Since, D is the midpoint of AB AD=DB =>\(\frac\) By converse of Basic Proportionality Theorem, DE || BC Hence, proved. NCERT solutions of related questions for Triangles • In figure. (i) and (ii), DE || BC. Find EC in (i) and AD in (ii). • E and F are points on the sides PQ and PR respectively of a \(\triangle PQR\). For each of the following cases, state whether EF || QR. (i) PE = 3.9 cm, EQ = 3 cm, PF = 3.6 cm and FR = 2.4 cm (ii) PE = 4 cm, QE = 4.5 cm, PF = 8 cm and RF = 9 cm (iii) PQ = 1.28 cm, PR = 2.56 cm, PE = 0.18 cm and PF = 0.63 cm • In the figure, if LM || CB and LN || CD, prove that \(\frac\). Show that ABCD is a trapezium.