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MATHEMATICS [3 marks sum]

Mid - point Theorem and Intercept Theorem · ICSE STD 9 (ICSE - English Medium). 20 questions.

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[3 marks sum]

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20 questions · timed · auto-graded

Question 13 Marks
In the adjoining figure, ABCD is a parallelogram, E is the mid-point of CD and through D, a line is drawn parallel to EB to meet CB produced at G and intersecting AB at F .
Prove that:
(i) $AD =\frac{1}{2} GC$
(ii) $DG =2 EB$.
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Answer
self
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Question 23 Marks
If $D , E , F$ are respectively the mid-points of the sides $AB , BC$ and CA of an equilateral triangle ABC , prove that $\triangle DEF$ is also an equilateral triangle.
Answer
self
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Question 33 Marks
In the given figure, D, E, F are respectively the midpoints of the sides $AB , BC$ and CA of $\triangle ABC$. Prove that ADEF is a parallelogram.
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Answer
self
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Question 43 Marks
Prove that the straight lines joining the mid-points of the opposite sides of a quadrilateral bisect each other.
Answer
[Hint. The quadrilateral formed by joining the mid-points of the pairs of adjacent sides of a quadrilateral is a $\| gm$. These lines will be the diagonals of that $\| gm$ and the diagonals of a $\| gm$ bisect each other.]
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Question 53 Marks
In the adjoining figure, ABCD is a trapezium in which $AB \| DC$ and E is the mid-point of AD . If $EF \| AB$ meets BC at F , show that F is the mid-point of BC .
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Answer
[Hint. $A B\|E F\| D C$ and $A D$ is the transversal such that $A E=E D$. Another transversal $B F C$ cuts them at $B, F$ and $C$ respectively. So, $B F=F C$.]
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Question 63 Marks
In the adjoining figure, ABCD is a trapezium in which $AB \| DC$. If M and N are the mid-points of AC and BD respectively, prove that $MN =\frac{1}{2}( AB - CD )$.
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Answer
[Hint. Join $C N$ and produce it to meet $A B$ at $E$.
Then, $\triangle C D N \equiv \triangle E B N$. So, $C D=E B$ and $C N=N E$.
$\left.M N=\frac{1}{2} A E(w h y ?)=\frac{1}{2}(A B-E B)=\frac{1}{2}(A B-C D) .\right]$
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Question 73 Marks
Show that the quadrilateral formed by joining the mid-points of the pairs of adjacent sides of a square is a square.
Answer
self
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Question 83 Marks
Show that the quadrilateral formed by joining the midpoints of the pairs of adjacent sides of a rhombus is a rectangle.
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Answer
[Hint. Join AC and BD to intersect at $O$.
Show that PQRS is a $\|$ gm.
Now, EOFR is a $\| gm$. So, $\angle E R F=\angle E O F=90$.
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Question 93 Marks
Show that the quadrilateral formed by joining the mid-points of the pairs of adjacent sides of a rectangle is a rhombus.
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Answer
[Hint. Join $A C$. Then $P Q \| A C$ and $P Q=\frac{1}{2} A C$. Also $S R \| A C$ and $S R=\frac{1}{2} A C$.
$\therefore$ PQRS is a $\| gm$.
Now, $\triangle A S P \equiv \triangle B Q P$. So, $P S=P Q$.
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Question 103 Marks
In the adjoining figure, $\triangle ABC$ is right-angled at B and P is the mid-point of AC . Show that, $PA = PB = PC$.
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Answer
[Hint. Draw $P Q \| B C$. Then, $Q$ is mid-point of $A B$ (why ?) Show that $\triangle A Q P \cong \triangle B Q P$ and therefore, $P A=P B$.]
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Question 113 Marks
In the given figure, D, E, F are the mid-points of the sides $BC , CA$ and AB respectively.
(i) If $AB =6.2 cm$, find DE .
(ii) If $DF =3.8 cm$, find AC .
(iii) If perimeter of $\triangle ABC$ is 21 cm , find FE.
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Answer
(i) $DE =3.1 cm$
(ii) $AC =7.6 cm$
(iii) $FE =3.6 cm$
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Question 123 Marks
Prove that the straight lines joining the mid-points of the opposite sides of a quadrilateral bisect each other.
Answer
self
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Question 133 Marks
In the adjoining figure, $A B C D$ is a trapezium in which $A B \| D C$ and $E$ is the mid-point of $A D$. If $E F \| A B$ meets $B C$ at $F$, show that $F$ is the mid-point of $B C$.
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Answer
self
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Question 143 Marks
In the adjoining figure, ABCD is a trapezium in which $A B \| D C$. If $M$ and $N$ are the mid-points of $A C$ and $B D$ respectively, prove that $MN =\frac{1}{2}( AB - CD )$.
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Answer
self
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Question 153 Marks
Show that the quadrilateral formed by joining the midpoints of the pairs of adjacent sides of a rhombus is a rectangle.
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Answer
self
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Question 163 Marks
In the adjoining figure, $\triangle ABC$ is right-angled at B and P is the mid-point of AC . Show that, $PA = PB = PC$.
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Answer
self
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Question 173 Marks
In the adjoining figure, ABCD is a parallelogram, E is the mid-point of $C D$ and through $D$, a line is drawn parallel to EB to meet CB produced at G and intersecting AB at F .
Prove that: (i) $AD =\frac{1}{2} GC \quad$ (ii) $DG =2 EB$.
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Answer
self
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Question 183 Marks
If $D , E , F$ are respectively the mid-points of the sides $AB , BC$ and CA of an equilateral triangle ABC , prove that $\triangle DEF$ is also an equilateral triangle.
Answer
self
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Question 193 Marks
In the given figure, $D , E , F$ are respectively the midpoints of the sides $A B, B C$ and $C A$ of $\triangle A B C$. Prove that ADEF is a parallelogram.
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Answer
self
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Question 203 Marks
In the given figure, $D , E , F$ are the mid-points of the sides $B C, C A$ and $A B$ respectively.
(i) If $AB =6.2 cm$, find DE .
(ii) If $DF =3.8 cm$, find AC .
(iii) If perimeter of $\triangle A B C$ is 21 cm , find FE .
Image
Answer
(i) $DE =3.1 cm$
(ii) $AC =7.6 cm$
(iii) $FE =3.6 cm$
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[3 marks sum] - MATHEMATICS STD 9 Questions - Vidyadip