[4 marks sum]

Question 14 Marks

In the adjoining figure, in $\triangle ABC , AD$ is the median through A and E is the mid-point of AD . If BE produced meets AC in F , prove that $AF =\frac{1}{3} AC$.

Answer

[Hint. Draw $D G \| B F$, meeting $A C$ at $G$].

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Question 24 Marks

In the adjoining figure, ABCD is a quadrilateral in which $AD = BC$ and $P , Q , R , S$ are the mid-points of $AB , BD , CD$ and $A C$ respectively. Prove that $P Q R S$ is a rhombus.

Answer

[Hint. $P Q \| A D$ and $P Q=\frac{1}{2} A D$.
Also, $S R \| A D$ and $S R=\frac{1}{2} A D$.
Also, $Q R=\frac{1}{2} B C=\frac{1}{2} A D=S R$.]

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Question 34 Marks

Two points A and B lie on the same side of a line XY . If $AD \perp XY$ and $BE \perp XY$ meet XY in D and E respectively and $C$ is the mid-point of $A B$, show that $CD = CE$.

Answer

[Hint. Draw $C F \perp X Y$. Then, $A D\|C F\| B E$ and $C$ is the mid-point of $A B$, so $F$ is the mid-point of $D E$. Now, show that $\triangle C F D \equiv \triangle C F E$].

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Question 44 Marks

Two points $A$ and $B$ lie on the same side of a line $X Y$. If $AD \perp XY$ and $BE \perp XY$ meet XY in D and E respectively and $C$ is the mid-point of $A B$, show that $CD = CE$.

Answer

self

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Question 54 Marks

In the adjoining figure, in $\triangle ABC , AD$ is the median through A and E is the mid-point of AD . If BE produced meets AC in F , prove that $AF =\frac{1}{3} AC$.

Answer

self

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Question 64 Marks

In the adjoining figure, ABCD is a quadrilateral in which $A D=B C$ and $P, Q, R, S$ are the mid-points of $A B, B D, C D$ and $A C$ respectively. Prove that $P Q R S$ is a rhombus.

Answer

self

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MATHEMATICS [4 marks sum]

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