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Standard IX

Mathematics

Every Point on the Bisector of an Angle Is Equidistant from the Sides of the Angle.

CD and GH a...

Question

$C D$ and $G H$ are respectively the bisectors of $∠ A C B$ and $∠ E G F$ such that $D$ and $H$ lie on the sides $A B$ and $F E$ of $△ A B C$ and $△ E F G$ . If $△ A B C \sim △ E F G$ , show that, $\frac{C D}{G H} = \frac{A C}{F G}$ .

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Solution

given $Δ A B C \sim Δ E F G$
$∠ A$ $= ∠ E$ , $∠ B$ $= ∠ F$ , $∠ C$ $= ∠ G$
$∠ A$ $= ∠ E$
$⟹ \frac{1}{2}$ $∠ C =$ $\frac{1}{2} ∠ G$
in triangle $Δ A D C$ and $Δ F G H$
we have $∠ A = ∠ F$ and $∠ A C D = ∠ H G F$
applying triangle sum properties we have
$∠ A D C = ∠ F G H$
$Δ A D G \sim Δ F H G$
$\frac{C D}{G H}$ = $\frac{A C}{F G}$

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Similar questions

C D

and

G H

are respectively the bisectors of

∠ A C B

and

∠ E G F

such that

D

and

H

lie on sides

A B

and

F E

△ A B C

and

△ E F G

respectively. If

△ A B C \sim △ F E G

, show that:
(i)

\frac{C D}{G H} = \frac{A C}{F G}

(ii)

△ D C B \sim △ H G E

(iii)

△ D C A \sim △ H G F

Q. Question 10 (i)
CD and GH are respectively the bisectors of

∠ A C B a n d ∠ E G F

such that D and H lie on sides AB and FE of

Δ A B C

and

Δ E F G

respectively. If

Δ A B C \sim Δ F E G

, Show that

(i) \frac{C D}{G H} = \frac{A C}{F G}

Q. Question 10 (i)
CD and GH are respectively the bisectors of

∠ A C B a n d ∠ E G F

such that D and H lie on sides AB and FE of

Δ A B C

and

Δ E F G

respectively. If

Δ A B C \sim Δ F E G

, Show that

(i) \frac{C D}{G H} = \frac{A C}{F G}

CD and GH are respectively the bisectors of ∠ACB and ∠EGF such that D and H lie on sides AB and FE of ΔABC and ΔEFG respectively. If ΔABC ∼ ΔFEG, Show that:

(i)

(ii) ΔDCB ∼ ΔHGE

(iii) ΔDCA ∼ ΔHGF

Q. Question 10 (iii)
CD and GH are respectively the bisectors of

a n g l e A C B

and

∠ E G F

such that D and H lie on sides AB and FE of

Δ A B C

and

Δ E F G

respectively. If

Δ A B C \sim Δ F E G

, Show that

(i i i) Δ D C A \sim Δ H G F

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CD and GH are respectively the bisectors of ∠ACB and ∠EGF such that D and H lie on the sides AB and FE of △ABC and △EFG. If △ABC∼△EFG, show that,CDGH=ACFG.

given ΔABC∼ΔEFG∠A =∠E, ∠B =∠F,∠C =∠G∠A =∠E⟹12 ∠C= 12∠Gin triangle ΔADC and ΔFGH we have ∠A=∠F and ∠ACD=∠HGFapplying triangle sum properties we have∠ADC=∠FGHΔADG∼ΔFHGCDGH =ACFG

$C D$ and $G H$ are respectively the bisectors of $∠ A C B$ and $∠ E G F$ such that $D$ and $H$ lie on the sides $A B$ and $F E$ of $△ A B C$ and $△ E F G$ . If $△ A B C \sim △ E F G$ , show that, $\frac{C D}{G H} = \frac{A C}{F G}$ .