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Class 9 Math Ratio and Proportion 4. Prove that the line segment drawn through the mid-point of one side of a triangle and parallel to another side bisects the third side.


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4. Prove that the line segment drawn through the mid-point of one side of a triangle and parallel to another side bisects the third side.

1. In \triangle \mathrm{ABC} \overline{\mathrm{DE}} \| \overline{\mathrm{BC}} .(iv) If \mathrm{m} \overline{\mathrm{AD}}=2.4 \mathrm{~cm} \mathrm{~m} \overline{\mathrm{AE}}=3.2 \mathrm{~cm} \mathrm{~m} \overline{\mathrm{DE}}=2 \mathrm{~cm} \mathrm{~m} \overline{\mathrm{BC}}=5 \mathrm{~cm} find \mathrm{m} \overline{\mathrm{AB}} \mathrm{m} \overline{\mathrm{DB}} \mathrm{m} \overline{\mathrm{AC}} \mathrm{m} \overline{\mathrm{CE}} .
1. In  \triangle \mathrm{ABC} \overline{\mathrm{DE}} \| \overline{\mathrm{BC}} .(iv) If  \mathrm{m} \overline{\mathrm{AD}}=2.4 \mathrm{~cm} \mathrm{~m} \overline{\mathrm{AE}}=3.2 \mathrm{~cm} \mathrm{~m} \overline{\mathrm{DE}}=2 \mathrm{~cm} \mathrm{~m} \overline{\mathrm{BC}}=5 \mathrm{~cm}  find  \mathrm{m} \overline{\mathrm{AB}}   \mathrm{m} \overline{\mathrm{DB}} \mathrm{m} \overline{\mathrm{AC}} \mathrm{m} \overline{\mathrm{CE}} .

1. In \triangle \mathrm{ABC} \overline{\mathrm{DE}} \| \overline{\mathrm{BC}} .(iv) If \mathrm{m} \overline{\mathrm{AD}}=2.4 \mathrm{~cm} \mathrm{~m} \overline{\mathrm{AE}}=3.2 \mathrm{~cm} \mathrm{~m} \overline{\mathrm{DE}}=2 \mathrm{~cm} \mathrm{~m} \overline{\mathrm{BC}}=5 \mathrm{~cm} find \mathrm{m} \overline{\mathrm{AB}} \mathrm{m} \overline{\mathrm{DB}} \mathrm{m} \overline{\mathrm{AC}} \mathrm{m} \overline{\mathrm{CE}} .

2. In \triangle \mathrm{ABC} shown in the figure \overrightarrow{\mathrm{CD}} bisects \angle \mathrm{C} . If \mathrm{m} \overline{\mathrm{AC}}=3 \mathrm{~m} \overline{\mathrm{CB}}=6 and m \overline{A B}=7 then find m \overline{A D} and
2. In  \triangle \mathrm{ABC}  shown in the figure  \overrightarrow{\mathrm{CD}}  bisects  \angle \mathrm{C} . If  \mathrm{m} \overline{\mathrm{AC}}=3 \mathrm{~m} \overline{\mathrm{CB}}=6  and  m \overline{A B}=7  then find  m \overline{A D}  and

2. In \triangle \mathrm{ABC} shown in the figure \overrightarrow{\mathrm{CD}} bisects \angle \mathrm{C} . If \mathrm{m} \overline{\mathrm{AC}}=3 \mathrm{~m} \overline{\mathrm{CB}}=6 and m \overline{A B}=7 then find m \overline{A D} and

4. Prove that the line segment drawn through the mid-point of one side of a triangle and parallel to another side bisects the third side.
4. Prove that the line segment drawn through the mid-point of one side of a triangle and parallel to another side bisects the third side.
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4. Prove that the line segment drawn through the mid-point of one side of a triangle and parallel to another side bisects the third side.

6. In Isosceles \triangle \mathrm{PQR} shown in the figure find the value of x and y .
6. In Isosceles  \triangle \mathrm{PQR}  shown in the figure find the value of  x  and  y .

6. In Isosceles \triangle \mathrm{PQR} shown in the figure find the value of x and y .

4. In the shown figure let \mathrm{m} \overline{\mathrm{PA}}=8 x-7 \mathrm{~m} \overline{\mathrm{PB}}=4 x-3 \mathrm{m} \overline{\mathrm{AQ}}=5 x-3 \mathrm{~m} \overline{\mathrm{BR}}=3 x-1 . Find the value of x if \overline{\mathrm{AB}} \| \overline{\mathrm{QR}} .
4. In the shown figure let  \mathrm{m} \overline{\mathrm{PA}}=8 x-7 \mathrm{~m} \overline{\mathrm{PB}}=4 x-3   \mathrm{m} \overline{\mathrm{AQ}}=5 x-3 \mathrm{~m} \overline{\mathrm{BR}}=3 x-1 .  Find the value of  x  if  \overline{\mathrm{AB}} \| \overline{\mathrm{QR}} .

4. In the shown figure let \mathrm{m} \overline{\mathrm{PA}}=8 x-7 \mathrm{~m} \overline{\mathrm{PB}}=4 x-3 \mathrm{m} \overline{\mathrm{AQ}}=5 x-3 \mathrm{~m} \overline{\mathrm{BR}}=3 x-1 . Find the value of x if \overline{\mathrm{AB}} \| \overline{\mathrm{QR}} .

5. Prove that the line segment joining the mid-points of any two sides of a triangle is parallel to the third side.
5. Prove that the line segment joining the mid-points of any two sides of a triangle is parallel to the third side.

5. Prove that the line segment joining the mid-points of any two sides of a triangle is parallel to the third side.

3. Show that in any correspondence of two triangles if two angles of one triangle are congruent to the corresponding angles of the other then the triangles are similar.
3. Show that in any correspondence of two triangles if two angles of one triangle are congruent to the corresponding angles of the other then the triangles are similar.

3. Show that in any correspondence of two triangles if two angles of one triangle are congruent to the corresponding angles of the other then the triangles are similar.

3. In an equilateral triangle \mathrm{ABC} shown in the figure \mathrm{m} \overline{\mathrm{AE}}: \mathrm{m} \overline{\mathrm{AC}}=\mathrm{m} \overline{\mathrm{AD}}: \mathrm{m} \overline{\mathrm{AB}} Find all the three angles of \triangle \mathrm{ADE} and name it also.
3. In an equilateral triangle  \mathrm{ABC}  shown in the figure  \mathrm{m} \overline{\mathrm{AE}}: \mathrm{m} \overline{\mathrm{AC}}=\mathrm{m} \overline{\mathrm{AD}}: \mathrm{m} \overline{\mathrm{AB}}  Find all the three angles of  \triangle \mathrm{ADE}  and name it also.

3. In an equilateral triangle \mathrm{ABC} shown in the figure \mathrm{m} \overline{\mathrm{AE}}: \mathrm{m} \overline{\mathrm{AC}}=\mathrm{m} \overline{\mathrm{AD}}: \mathrm{m} \overline{\mathrm{AB}} Find all the three angles of \triangle \mathrm{ADE} and name it also.

5. In \Delta \mathrm{LMN} shown in the figure \overrightarrow{\mathrm{LA}} bisects \angle \mathrm{L} . If \mathrm{mLN}=4 \mathrm{mLM}=6 \mathrm{mMN}=8 then
5. In  \Delta \mathrm{LMN}  shown in the figure  \overrightarrow{\mathrm{LA}}  bisects  \angle \mathrm{L} . If  \mathrm{mLN}=4 \mathrm{mLM}=6 \mathrm{mMN}=8  then

5. In \Delta \mathrm{LMN} shown in the figure \overrightarrow{\mathrm{LA}} bisects \angle \mathrm{L} . If \mathrm{mLN}=4 \mathrm{mLM}=6 \mathrm{mMN}=8 then

1. In \triangle \mathrm{ABC} \overline{\mathrm{DE}} \| \overline{\mathrm{BC}} .(i) If \mathrm{m} \overline{\mathrm{AD}}=1.5 \mathrm{~cm} \mathrm{~m} \overline{\mathrm{BD}}=3 \mathrm{~cm} \mathrm{~m} \overline{\mathrm{AE}}=1.3 \mathrm{~cm} then find \mathrm{m} \overline{\mathrm{CE}} .
1. In  \triangle \mathrm{ABC} \overline{\mathrm{DE}} \| \overline{\mathrm{BC}} .(i) If  \mathrm{m} \overline{\mathrm{AD}}=1.5 \mathrm{~cm} \mathrm{~m} \overline{\mathrm{BD}}=3 \mathrm{~cm} \mathrm{~m} \overline{\mathrm{AE}}=1.3 \mathrm{~cm}  then find  \mathrm{m} \overline{\mathrm{CE}} .

1. In \triangle \mathrm{ABC} \overline{\mathrm{DE}} \| \overline{\mathrm{BC}} .(i) If \mathrm{m} \overline{\mathrm{AD}}=1.5 \mathrm{~cm} \mathrm{~m} \overline{\mathrm{BD}}=3 \mathrm{~cm} \mathrm{~m} \overline{\mathrm{AE}}=1.3 \mathrm{~cm} then find \mathrm{m} \overline{\mathrm{CE}} .

1. Which of the following are true and which are false?(i) Congruent triangles are of same size and shape.(ii) Similar triangles are of same shape but different sizes.(iii) Symbol used for congruent is \cong.(iv) Symbol used for similarity is (v) Congruent triangles are similar.(vi). Similar triangles are congruent.(vii) A line segment has only one mid-point.(viii) One and only one line can be drawn through two points.(ix) Proportion is non-equality of two ratios.(x) Ratio has no unit.
1. Which of the following are true and which are false?(i) Congruent triangles are of same size and shape.(ii) Similar triangles are of same shape but different sizes.(iii) Symbol used for congruent is \cong.(iv) Symbol used for similarity is  (v) Congruent triangles are similar.(vi). Similar triangles are congruent.(vii) A line segment has only one mid-point.(viii) One and only one line can be drawn through two points.(ix) Proportion is non-equality of two ratios.(x) Ratio has no unit.

1. Which of the following are true and which are false?(i) Congruent triangles are of same size and shape.(ii) Similar triangles are of same shape but different sizes.(iii) Symbol used for congruent is \cong.(iv) Symbol used for similarity is (v) Congruent triangles are similar.(vi). Similar triangles are congruent.(vii) A line segment has only one mid-point.(viii) One and only one line can be drawn through two points.(ix) Proportion is non-equality of two ratios.(x) Ratio has no unit.

1. In \triangle \mathrm{ABC} \overline{\mathrm{DE}} \| \overline{\mathrm{BC}} .(v) If \mathrm{AD}=4 x-3 \mathrm{AE}=8 x-7 \mathrm{BD}=3 x-1 and \mathrm{CE}=5 x-3 find the value of x .
1. In  \triangle \mathrm{ABC} \overline{\mathrm{DE}} \| \overline{\mathrm{BC}} .(v) If  \mathrm{AD}=4 x-3 \mathrm{AE}=8 x-7 \mathrm{BD}=3 x-1  and  \mathrm{CE}=5 x-3  find the value of  x .

1. In \triangle \mathrm{ABC} \overline{\mathrm{DE}} \| \overline{\mathrm{BC}} .(v) If \mathrm{AD}=4 x-3 \mathrm{AE}=8 x-7 \mathrm{BD}=3 x-1 and \mathrm{CE}=5 x-3 find the value of x .

1. In \triangle \mathrm{ABC} \overline{\mathrm{DE}} \| \overline{\mathrm{BC}} .(ii) If \mathrm{m} \overline{\mathrm{AD}}=2.4 \mathrm{~cm} \mathrm{~m} \overline{\mathrm{AE}}=3.2 \mathrm{~cm} \mathrm{~m} \overline{\mathrm{EC}}=4.8 \mathrm{~cm} find \mathrm{mAB} .
1. In  \triangle \mathrm{ABC} \overline{\mathrm{DE}} \| \overline{\mathrm{BC}} .(ii) If  \mathrm{m} \overline{\mathrm{AD}}=2.4 \mathrm{~cm} \mathrm{~m} \overline{\mathrm{AE}}=3.2 \mathrm{~cm} \mathrm{~m} \overline{\mathrm{EC}}=4.8 \mathrm{~cm}  find  \mathrm{mAB} .

1. In \triangle \mathrm{ABC} \overline{\mathrm{DE}} \| \overline{\mathrm{BC}} .(ii) If \mathrm{m} \overline{\mathrm{AD}}=2.4 \mathrm{~cm} \mathrm{~m} \overline{\mathrm{AE}}=3.2 \mathrm{~cm} \mathrm{~m} \overline{\mathrm{EC}}=4.8 \mathrm{~cm} find \mathrm{mAB} .

2. If \triangle \mathrm{ABC} is an isosceles triangle \angle \mathrm{A} is vertex angle and \overline{\mathrm{DE}} intersects the sides \overline{\mathrm{AB}} and \overline{\mathrm{AC}} as shown in the figure so that
2. If  \triangle \mathrm{ABC}  is an isosceles triangle  \angle \mathrm{A}  is vertex angle and  \overline{\mathrm{DE}}  intersects the sides  \overline{\mathrm{AB}}  and  \overline{\mathrm{AC}}  as shown in the figure so that

2. If \triangle \mathrm{ABC} is an isosceles triangle \angle \mathrm{A} is vertex angle and \overline{\mathrm{DE}} intersects the sides \overline{\mathrm{AB}} and \overline{\mathrm{AC}} as shown in the figure so that

4. If line segments A B and C D are intersecting at point X and \frac{m \overline{A X}}{m \overline{X B}}=\frac{m \overline{C X}}{m \overline{X D}} then show that \triangle \mathrm{AXC} and \Delta \mathrm{BXD} are similar.
4. If line segments  A B  and  C D  are intersecting at point  X  and  \frac{m \overline{A X}}{m \overline{X B}}=\frac{m \overline{C X}}{m \overline{X D}}  then show that  \triangle \mathrm{AXC}  and  \Delta \mathrm{BXD}  are similar.

4. If line segments A B and C D are intersecting at point X and \frac{m \overline{A X}}{m \overline{X B}}=\frac{m \overline{C X}}{m \overline{X D}} then show that \triangle \mathrm{AXC} and \Delta \mathrm{BXD} are similar.

3. In \Delta \mathrm{LMN} shown in the figure \overline{\mathrm{MN}} \| \overline{\mathrm{PQ}} (i) If \mathrm{mLM}=5 \mathrm{~cm} \mathrm{mLP}=2.5 \mathrm{~cm} \mathrm{mLQ}=2.3 \mathrm{~cm} then find \mathrm{mLN} .(ii) If \mathrm{mLM}=6 \mathrm{~cm} \mathrm{mLQ}=2.5 \mathrm{~cm} \mathrm{~m} \overline{\mathrm{QN}}=5 \mathrm{~cm}
3. In  \Delta \mathrm{LMN}  shown in the figure  \overline{\mathrm{MN}} \| \overline{\mathrm{PQ}} (i) If  \mathrm{mLM}=5 \mathrm{~cm} \mathrm{mLP}=2.5 \mathrm{~cm} \mathrm{mLQ}=2.3 \mathrm{~cm}  then find  \mathrm{mLN} .(ii) If  \mathrm{mLM}=6 \mathrm{~cm} \mathrm{mLQ}=2.5 \mathrm{~cm} \mathrm{~m} \overline{\mathrm{QN}}=5 \mathrm{~cm}

3. In \Delta \mathrm{LMN} shown in the figure \overline{\mathrm{MN}} \| \overline{\mathrm{PQ}} (i) If \mathrm{mLM}=5 \mathrm{~cm} \mathrm{mLP}=2.5 \mathrm{~cm} \mathrm{mLQ}=2.3 \mathrm{~cm} then find \mathrm{mLN} .(ii) If \mathrm{mLM}=6 \mathrm{~cm} \mathrm{mLQ}=2.5 \mathrm{~cm} \mathrm{~m} \overline{\mathrm{QN}}=5 \mathrm{~cm}

1. In \triangle \mathrm{ABC} as shown in the figure \overrightarrow{\mathrm{CD}} bisects \angle \mathrm{C} and meets \overline{\mathrm{AB}} at \mathrm{D} . \mathrm{mBD} is equal to \begin{array}{llll}\text { (a) } 5 & \text { (b) } 16 & \text { (c) } 10 & \text { (d) } 18 & \end{array}
1. In  \triangle \mathrm{ABC}  as shown in the figure  \overrightarrow{\mathrm{CD}}  bisects  \angle \mathrm{C}  and meets  \overline{\mathrm{AB}}  at  \mathrm{D} . \mathrm{mBD}  is equal to  \begin{array}{llll}\text { (a) } 5 & \text { (b) } 16 & \text { (c) } 10 & \text { (d) } 18 & \end{array}

1. In \triangle \mathrm{ABC} as shown in the figure \overrightarrow{\mathrm{CD}} bisects \angle \mathrm{C} and meets \overline{\mathrm{AB}} at \mathrm{D} . \mathrm{mBD} is equal to \begin{array}{llll}\text { (a) } 5 & \text { (b) } 16 & \text { (c) } 10 & \text { (d) } 18 & \end{array}

1. In \triangle \mathrm{ABC} \overline{\mathrm{DE}} \| \overline{\mathrm{BC}} .(iii) If \frac{\mathrm{m} \overline{\mathrm{AD}}}{\mathrm{m} \overline{\mathrm{DB}}}=\frac{3}{5} and \mathrm{m} \overline{\mathrm{AC}}=4.8 \mathrm{~cm} find \mathrm{mAE} .
1. In  \triangle \mathrm{ABC} \overline{\mathrm{DE}} \| \overline{\mathrm{BC}} .(iii) If  \frac{\mathrm{m} \overline{\mathrm{AD}}}{\mathrm{m} \overline{\mathrm{DB}}}=\frac{3}{5}  and  \mathrm{m} \overline{\mathrm{AC}}=4.8 \mathrm{~cm}  find  \mathrm{mAE} .

1. In \triangle \mathrm{ABC} \overline{\mathrm{DE}} \| \overline{\mathrm{BC}} .(iii) If \frac{\mathrm{m} \overline{\mathrm{AD}}}{\mathrm{m} \overline{\mathrm{DB}}}=\frac{3}{5} and \mathrm{m} \overline{\mathrm{AC}}=4.8 \mathrm{~cm} find \mathrm{mAE} .

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