In a triangle ABC, AB = AC and D is a point on side AC such that BC² = AC × CD.
Prove that BD = BC.

Question

In a triangle ABC, AB = AC and D is a point on side AC such that BC² = AC × CD. Prove that BD = BC.

KnowledgeBoat · Accepted Answer

Draw BE ⊥ AC. In right triangle BEC, By pythagoras theorem, ⇒ BC² = BE² + EC² ⇒ BC² = BE² + (AC - AE)² ⇒ BC² = BE² + AC² + AE² - 2AC.AE .........(i) In right triangle ABE, By pythagoras theorem, ⇒ AB² = BE² + AE².........(ii) Substituting value of BE² + AE² from (ii) in (i) we get, ⇒ BC² = AB² + AC² - 2AC.CE ⇒ BC² = AC² + AC² - 2AC.AE [∵ AB = AC] ⇒ BC² = 2AC² - 2AC.AE ⇒ BC² = 2AC(AC - AE) ⇒ BC² = 2AC × EC .........(iii) Given, BC² = AC × CD ..........(iv) Comparing (iii) and (iv) we get, ⇒ 2AC × EC = AC × CD ⇒ 2EC = CD or EC = So, we can say that ED = EC as E is mid-point of CD. In △BDE and △BCE, BE = BE (Common) ED = EC ∠BED = ∠BEC = 90° △BDE ≅ △BCE by SAS axiom of congruency. We know that corresponding parts of congruent triangle are equal. ∴ BD = BC. Hence, proved that BD = BC.

Mathematics

In a triangle ABC, AB = AC and D is a point on side AC such that BC² = AC × CD.
Prove that BD = BC.

Pythagoras Theorem

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Mathematics

In a triangle ABC, AB = AC and D is a point on side AC such that BC2 = AC × CD.Prove that BD = BC.

Pythagoras Theorem

Related Questions

In the adjoining figure, △PQR is right angled at Q and points S and T trisect side QR. Prove that8PT2 = 3PR2 + 5PS2.

In a quadrilateral ABCD, ∠B = 90°. If AD2 = AB2 + BC2 + CD2, Prove that ∠ACD = 90°.

In the adjoining figure, find the length of AD in terms of b and c.

ABCD is a square, F is mid-point of AB and BE is one third of BC. If area of △FBE is 108 cm2, find the length of AC.

In a triangle ABC, AB = AC and D is a point on side AC such that BC² = AC × CD.
Prove that BD = BC.

In the adjoining figure, △PQR is right angled at Q and points S and T trisect side QR. Prove that
8PT² = 3PR² + 5PS².

In a quadrilateral ABCD, ∠B = 90°. If AD² = AB² + BC² + CD², Prove that ∠ACD = 90°.

ABCD is a square, F is mid-point of AB and BE is one third of BC. If area of △FBE is 108 cm², find the length of AC.