In Fig 2, D E ∥ A C and D C ∥ A P . Prove that B E E C = B C C P

In the given figure, it is given that D E ∥ A C and D C ∥ A P We know that, 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. In ∆ B A C , as D E ∥ A C , According to the basic proportionality theorem as stated above, B D A D = B E E C .....(i) In ∆ B A P , as D C ∥ A P , According to the basic proportionality theorem, B D A D = B C C P .....(ii) From (i) and (ii), we get B E E C = B C C P Hence, proved.

A B C D is a cyclic quadrilateral. Prove that A C × B D = A B × D C + A D × B C

Take a point on M on B D so that ∠ A C B = ∠ M C D As we know that the angles in the same segment are equal. Then, ∠ B A C = ∠ B D C . Now, by A A similarity, we have △ A C B ~ △ D C B . Thus, we get A C C D = A B D M ⇒ A B × C D = A C × D M . . . . . . ( 1 ) Now, ∠ D A C = ∠ D B C and ∠ D C A = ∠ B C M , then by A A similarity, we have △ A C D ~ △ B C M . Thus, we get A C B C = A D B M ⇒ A D × B C = A C × B M . . . . . ( 2 ) Add equation ( 1 ) and ( 2 ) A C × D M + A C × B M = A B × D C + A D × B C A C ( D M + B M ) = A B × D C + A D × B C A C × B D = A B × D C + A D × B C Hence, proved

In given figure, ∠ A C B = 90 ° and C D ⊥ A B , prove that C D 2 = B D × A D

Given, in ∆ A B C , ∠ A C B = 90 ° and C D ⊥ A B . To prove: C D 2 = B D × A D Proof: In ∆ C A D , C A 2 = C D 2 + A D 2 ------------(i) Similarly, In ∆ C D B , C B 2 = C D 2 + B D 2 -------(ii) Adding (i) and (ii), we get, C A 2 + C B 2 = C D 2 + A D 2 + C D 2 + B D 2 ⇒ A B 2 = 2 C D 2 + A D 2 + B D 2 ⇒ A B 2 - A D 2 = B D 2 + 2 C D 2 ⇒ A B + A D A B - A D - B D 2 = 2 C D 2 ⇒ A B + A D B D - B D 2 = 2 C D 2 ⇒ B D A B + A D - B D = 2 C D 2 ⇒ B D A B - B D + A D = 2 C D 2 ⇒ B D × 2 A D = 2 C D 2 ⇒ C D 2 = B D × A D Hence, proved.

E M B I B E

Mathematics Crash Course (Based on Revised Syllabus-2023)>Chapter 2 - Triangles>Exercise

Embibe Experts Solutions for Chapter: Triangles, Exercise 1: Exercise

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Embibe Experts Mathematics Solutions for Exercise - Embibe Experts Solutions for Chapter: Triangles, Exercise 1: Exercise

Attempt the practice questions on Chapter 2: Triangles, Exercise 1: Exercise with hints and solutions to strengthen your understanding. Mathematics Crash Course (Based on Revised Syllabus-2023) solutions are prepared by Experienced Embibe Experts.