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Byju's Answer

Standard XII

Mathematics

Cosine Rule

In a triangle...

Question

In a triangle ABC, points X and Y are on AB and AC, respectively, such that XY is parallel to BC. Which of the two following always hold(s) good? (Here [PQR] denotes the area of triangle PQR.)

(I) [BCX] = [BCY]

(II) [ACX] .[ABY] = [AXY] .[ABC]

Neither [I] nor [II]

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[I] only

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[II] only

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both [I] and [II]

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Solution

The correct option is D
both [I] and [II]

$Δ B C X = Δ B C Y$ (Obvious)

Same base and same height

Now Let $A (\to O), \to A B = \to b, \to A C = \to c$

So $\to A X = λ \to b, \to A Y = λ \to c$

$Δ A C X = \frac{1}{2} λ | \to b \times \to c |$

$Δ A B Y = \frac{1}{2} λ | \to b \times \to c |$

$Δ A X Y = \frac{1}{2} λ^{2} | \to b \times \to c |$

$Δ A B C = \frac{1}{2} | \to b \times \to c |$

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Similar questions

Q. In a triangle

A B C

, points

X

and

Y

are on

A B

and

A C

respectively, such that

X Y

is parallel to

B C

. Which of the following always hold? (Here

[P Q R]

denotes the area of triangle

P Q R

)

(I)

[B C X] = [B C Y]

(I I)

[A C X] . [A B Y] = [A X Y] . [A B C]

$△$ ABC is an acute angled triangle. DE is drawn parallel to BC as shown. Which of the following are always true?

i) $△ A B C \sim △$ ADE

ii) $\frac{A D}{B D} = \frac{A E}{E C}$

iii) $D E = \frac{B C}{2}$

D and E are points on the sides AB and AC respectively of a $△$ ABC such that DE || BC. Which of the following statement is true?

(i) $△$ ADE ~ $△$ ABC

(ii) $\frac{a r e a o f △ A D E}{a r e a o f △ A B C}$ = $\frac{A D^{2}}{A B^{2}}$

(iii) $\frac{a r e a o f △ A D E}{a r e a o f △ A B C}$ = $\frac{A B^{2}}{A C^{2}}$

Q. In the figure, ABC is a triangle in which AB = AC. X and Y are points on AB and AC such that AX =AY. Then

Δ A B Y ≅ Δ A C X

Q. The incircle of a scalene triangle

A B C

touches side

A B, B C

and

A C

P, Q

and

R

respectively. If

O

is the incentre of triangle

A B C,

then which of the following statements is (are) correct?

I.

Area of

△ O B C

\frac{1}{2} \times B C \times radius of incircle

II.

A B + C Q = B Q + A C

III.

Area of

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\frac{1}{2} \times perimeter of △ A B C \times radius of incircle

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In a triangle ABC, points X and Y are on AB and AC, respectively, such that XY is parallel to BC. Which of the two following always hold(s) good? (Here [PQR] denotes the area of triangle PQR.) (I) [BCX] = [BCY] (II) [ACX] .[ABY] = [AXY] .[ABC]

The correct option is D both [I] and [II] ΔBCX=ΔBCY (Obvious) Same base and same height Now Let A(→O),→AB=→b,→AC=→c So →AX=λ→b,→AY=λ→c ΔACX=12λ|→b×→c| ΔABY=12λ|→b×→c| ΔAXY=12λ2|→b×→c| ΔABC=12|→b×→c|

In a triangle ABC, points X and Y are on AB and AC, respectively, such that XY is parallel to BC. Which of the two following always hold(s) good? (Here [PQR] denotes the area of triangle PQR.)

(I) [BCX] = [BCY]

(II) [ACX] .[ABY] = [AXY] .[ABC]