Question

If the given figure, ABC is a right triangle right angled at A, BCED, ACFGand ABMN are squares on the sides BC, CA and AB respectively. Line segment AX ⊥ DE meets BC at Y.

Show that
(i) Δ MBC $\cong$ Δ ABD

(ii) ar (BYXD) = 2 ar (Δ MBC)

(iii) ar (BYXD) = ar (ABMN)

(iv) Δ FCB $\cong$ Δ ACE

(v) ar (CYXE) = 2 ar (ΔFCB)

(vi) ar (CYXE) = ar (ACFG)

(vii) ar (BCED) = ar (ABMN) + ar (ACFG)

Answer 1

Given:

(1) ABCD is a right angled triangle at A

(2) BCED, ACFG and ABMN are the squares on the sides of BC, CA and AB respectively.

(3) , meets BC at Y.

To prove:

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

Proof:

(i)

…… (1)

(ii) Triangle ABD and rectangle BYXD are on the same base BD and between the same parallels AX and BD.

Therefore

(Using (1)) …… (2)

(iii) Since ΔMBC and square MBAN are on the same base MB and between the same parallels MB and NC.

…… (3)

From (2) and (3) we get

(iv) In triangle FCB and ACE

…… (4)

(v) Since ΔACE and rectangle CYXE are on the same base CE and between the same parallels CE and AX.

…… (5)

(vi) Since ΔFCB and rectangle FCAG are on the same base FC and between the same parallels FC and BG

…… (6)

From (5) and (6) we get

(vii) Applying Pythagoras Theorem in ΔACB, WE get