Question

AC.
АС
If D is the mid point of the hypotenuse AC of a right angled AABC. Prove that BD =
2
9)
E
o
B​

Answer 1

Answer:

GIVEN A ΔABC in which ∠B=90∘ and D is the midpoint of aC.

TO PROVE BD=12AC.

CONSTRUCTION Produce BD to E such that BD = DE.

Join EC.

PROOF In ΔADBandΔCDE, we have

AD=CD (given)

BD=ED (by construction)

and ∠ADE=∠CDE (vert. opp. ∠s).

∴ ΔADB≅ΔCDE (SAS-criteria).

∴ AB=ECand∠1=∠2 (c.p.c.t.).

But, ∠1and∠2 are alternate interior angles.

∴ CE||BA.

Now, CE||BA and BC is the transversal.

∴ ∠ABC+∠BCE=180∘ [co. int. ∠s]

⇒ 90∘+∠BCE=180∘ [∵∠ABC=90∘]

⇒ ∠BCE=90∘

Now, inΔABCandΔECB, we have

BC=CB (common),

AB=EC (proved)

and∠CBA=∠BCE (each equal to 90∘).

∴ ΔABC≅ΔECB (SAS-criteria).

∴ AC=EB⇒12EB=12AC⇒BD=12AC.

Hence, BD=12AC.

Explanation:

PLEASE MARKE AS BRAINLIST.