In the following figure- - AXY - - AYX- If BX-AX-CY-AY- show that triangle ABC is isosceles-

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Standard X

Mathematics

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Question

In the following figure, $∠$ AXY = $∠$ AYX. If $\frac{B X}{A X} = \frac{C Y}{A Y}$ , show that triangle ABC is isosceles.

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Solution

Given,
∠AXY = ∠AYX
⇒AX = AY

Also, BX/AX = CY/AY
⇒BX/AX + 1 = CY/AY + 1
⇒BX+AX/AX = CY+AY/AY
⇒AB/AX = AC/AY
⇒AB/AY = AC/AY [∵ AX = AY]
⇒AB = AC

Thus, the given triangle is isosceles.

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In the following figure, ∠ AXY = ∠ AYX. If BXAX=CYAY, show that triangle ABC is isosceles.

Given, ∠AXY = ∠AYX ⇒AX = AY Also, BX/AX = CY/AY ⇒BX/AX + 1 = CY/AY + 1 ⇒BX+AX/AX = CY+AY/AY ⇒AB/AX = AC/AY ⇒AB/AY = AC/AY [∵ AX = AY] ⇒AB = AC Thus, the given triangle is isosceles.

In the following figure, $∠$ AXY = $∠$ AYX. If $\frac{B X}{A X} = \frac{C Y}{A Y}$ , show that triangle ABC is isosceles.

Given,
∠AXY = ∠AYX
⇒AX = AY

Also, BX/AX = CY/AY
⇒BX/AX + 1 = CY/AY + 1
⇒BX+AX/AX = CY+AY/AY
⇒AB/AX = AC/AY
⇒AB/AY = AC/AY [∵ AX = AY]
⇒AB = AC

Thus, the given triangle is isosceles.