To divide a line segment $AB$ in the ratio $4:7,$ a ray $AX$ is drawn first such that $\angle BAX$ is an acute angle and then points $A_1, A_2, A_3, \dots$ are located at equal distances on the ray $AX$ and the point $B$ is joined to:

  • A
    $A_{12}$
  • B
    $A_{13}$
  • C
    $A_9$
  • D
    $A_{11}$

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Draw $\odot( P , 3 \, cm )$ and a diameter $\overline{ AB }$ in it. Take points $X$ and $Y$ such that $X - A - B$ and $A - B - Y$ where $PX = PY = 7 \, cm$. From $X$ and $Y$,draw tangents to $\odot( P , 3 \, cm )$. Write the steps of construction.

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Write True or False and give reasons for your answer.
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Draw $\overline{PQ}$ of length $8.5 \text{ cm}$ and obtain point $X$ on $\overline{PQ}$ such that $PX : XQ = 4 : 7$. Write the steps of construction.

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State whether the following statement is True or False and provide a reason for your answer:
$A$ pair of tangents can be constructed to a circle inclined at an angle of $170^{\circ}$.

Draw $\overline{ AB }$ of length $7 \,cm$ and obtain a point $M$ on $\overline{ AB }$ such that $AM : AB = 2 : 5$. Write the steps of construction.

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