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Class 10 Math Projection of a Side of a Triangle 6. In a triangle A B C m \overline{B C}=21 \mathrm{~cm} . m \overline{A C}=17 \mathrm{~cm} m \overline{A B}=10 \mathrm{~cm} . Calculate the projection of \overline{A B} upon \overline{B C} .


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6. In a triangle A B C m \overline{B C}=21 \mathrm{~cm} . m \overline{A C}=17 \mathrm{~cm} m \overline{A B}=10 \mathrm{~cm} . Calculate the projection of \overline{A B} upon \overline{B C} .

9. Whether the triangle with sides 5 \mathrm{~cm} 7 \mathrm{~cm} 8 \mathrm{~cm} is acute obtuce or right angled.
9. Whether the triangle with sides  5 \mathrm{~cm} 7 \mathrm{~cm} 8 \mathrm{~cm}  is acute obtuce or right angled.

9. Whether the triangle with sides 5 \mathrm{~cm} 7 \mathrm{~cm} 8 \mathrm{~cm} is acute obtuce or right angled.

6. In a triangle A B C m \overline{B C}=21 \mathrm{~cm} . m \overline{A C}=17 \mathrm{~cm} m \overline{A B}=10 \mathrm{~cm} . Calculate the projection of \overline{A B} upon \overline{B C} .
6. In a triangle  A B C m \overline{B C}=21 \mathrm{~cm} . m \overline{A C}=17 \mathrm{~cm} m \overline{A B}=10 \mathrm{~cm} . Calculate the projection of  \overline{A B}  upon  \overline{B C} .
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6. In a triangle A B C m \overline{B C}=21 \mathrm{~cm} . m \overline{A C}=17 \mathrm{~cm} m \overline{A B}=10 \mathrm{~cm} . Calculate the projection of \overline{A B} upon \overline{B C} .

8. In a \triangle A B C a=17 \mathrm{~cm} b=15 \mathrm{~cm} and c=8 \mathrm{~cm} find m \angle B .
8. In a  \triangle A B C a=17 \mathrm{~cm} b=15 \mathrm{~cm}  and  c=8 \mathrm{~cm}  find  m \angle B .

8. In a \triangle A B C a=17 \mathrm{~cm} b=15 \mathrm{~cm} and c=8 \mathrm{~cm} find m \angle B .

3. In a \triangle A B C calculate m \overline{B C} when m \overline{A B}=5 \mathrm{~cm} m \overline{A C}=4 \mathrm{~cm} m \angle A=60^{\circ} .
3. In a  \triangle A B C  calculate  m \overline{B C}  when  m \overline{A B}=5 \mathrm{~cm} m \overline{A C}=4 \mathrm{~cm} m \angle A=60^{\circ} .

3. In a \triangle A B C calculate m \overline{B C} when m \overline{A B}=5 \mathrm{~cm} m \overline{A C}=4 \mathrm{~cm} m \angle A=60^{\circ} .

4. In a \triangle A B C calculate m \overline{A C} when m \overline{A B}=5 \mathrm{~cm} m \overline{B C}=4 \sqrt{2} \mathrm{~cm} m \angle B=45^{\circ} .
4. In a  \triangle A B C  calculate  m \overline{A C}  when  m \overline{A B}=5 \mathrm{~cm} m \overline{B C}=4 \sqrt{2} \mathrm{~cm} m \angle B=45^{\circ} .

4. In a \triangle A B C calculate m \overline{A C} when m \overline{A B}=5 \mathrm{~cm} m \overline{B C}=4 \sqrt{2} \mathrm{~cm} m \angle B=45^{\circ} .

1. In a \triangle A B C m \angle A=60^{\circ} prove that (B C)^{2}=(A B)^{2}+(A C)^{2}-m \overline{A B} \cdot m \overline{A C} .
1. In a  \triangle A B C m \angle A=60^{\circ}  prove that  (B C)^{2}=(A B)^{2}+(A C)^{2}-m \overline{A B} \cdot m \overline{A C} .

1. In a \triangle A B C m \angle A=60^{\circ} prove that (B C)^{2}=(A B)^{2}+(A C)^{2}-m \overline{A B} \cdot m \overline{A C} .

2. Find m \overline{A C} if in \triangle A B C m \overline{B C}=6 \mathrm{~cm} m \overline{A B}=4 \sqrt{2} \mathrm{~cm} and m \angle A B C=135^{\circ} .
2. Find  m \overline{A C}  if in  \triangle A B C m \overline{B C}=6 \mathrm{~cm} m \overline{A B}=4 \sqrt{2} \mathrm{~cm}  and  m \angle A B C=135^{\circ} .

2. Find m \overline{A C} if in \triangle A B C m \overline{B C}=6 \mathrm{~cm} m \overline{A B}=4 \sqrt{2} \mathrm{~cm} and m \angle A B C=135^{\circ} .

10. Whether the triangle with sides 8 \mathrm{~cm} 15 \mathrm{~cm} 17 \mathrm{~cm} is acute obtuce or right angled.
10. Whether the triangle with sides  8 \mathrm{~cm} 15 \mathrm{~cm} 17 \mathrm{~cm}  is acute obtuce or right angled.

10. Whether the triangle with sides 8 \mathrm{~cm} 15 \mathrm{~cm} 17 \mathrm{~cm} is acute obtuce or right angled.

Example 1: In \triangle A B C \angle C is obtuse \overline{A D} \perp \overline{B C} produced whereas \overline{B D} is projection of \overline{A B} on \overline{B C} . Prove that (A C)^{2}=(A B)^{2}+(B C)^{2}-2 m \overline{B C} \cdot m \overline{B D}
Example 1: In  \triangle A B C \angle C  is obtuse  \overline{A D} \perp \overline{B C}  produced whereas  \overline{B D}  is projection of  \overline{A B}  on  \overline{B C} . Prove that  (A C)^{2}=(A B)^{2}+(B C)^{2}-2 m \overline{B C} \cdot m \overline{B D}

Example 1: In \triangle A B C \angle C is obtuse \overline{A D} \perp \overline{B C} produced whereas \overline{B D} is projection of \overline{A B} on \overline{B C} . Prove that (A C)^{2}=(A B)^{2}+(B C)^{2}-2 m \overline{B C} \cdot m \overline{B D}

2. In a \triangle A B C m \overline{A B}=6 \mathrm{~cm} m \overline{B C}=8 \mathrm{~cm} m \overline{A C}=9 \mathrm{~cm} and D is the mid point of side \overline{A C} . Find length of the median \overline{B D} .
2. In a  \triangle A B C m \overline{A B}=6 \mathrm{~cm} m \overline{B C}=8 \mathrm{~cm} m \overline{A C}=9 \mathrm{~cm}  and  D  is the mid point of side  \overline{A C} . Find length of the median  \overline{B D} .

2. In a \triangle A B C m \overline{A B}=6 \mathrm{~cm} m \overline{B C}=8 \mathrm{~cm} m \overline{A C}=9 \mathrm{~cm} and D is the mid point of side \overline{A C} . Find length of the median \overline{B D} .

1. In a \triangle A B C calculate m \overline{B C} when m \overline{A B}=6 \mathrm{~cm} m \overline{A C}=4 \mathrm{~cm} and m \angle A=60^{\circ} .
1. In a  \triangle A B C  calculate  m \overline{B C}  when  m \overline{A B}=6 \mathrm{~cm} m \overline{A C}=4 \mathrm{~cm}  and  m \angle A=60^{\circ} .

1. In a \triangle A B C calculate m \overline{B C} when m \overline{A B}=6 \mathrm{~cm} m \overline{A C}=4 \mathrm{~cm} and m \angle A=60^{\circ} .

Example 2: In an Isosceles \triangle A B C if m \overline{A B}=m \overline{A C} and \overline{B E} \perp \overline{A C} then prove that (B C)^{2}=2 m \overline{A C} \cdot m \overline{C E}
Example 2: In an Isosceles  \triangle A B C  if  m \overline{A B}=m \overline{A C}  and  \overline{B E} \perp \overline{A C}  then prove that  (B C)^{2}=2 m \overline{A C} \cdot m \overline{C E}

Example 2: In an Isosceles \triangle A B C if m \overline{A B}=m \overline{A C} and \overline{B E} \perp \overline{A C} then prove that (B C)^{2}=2 m \overline{A C} \cdot m \overline{C E}

3. In a parallelogram A B C D prove that (A C)^{2}+(B D)^{2}=2\left[(A B)^{2}+(B C)^{2}\right]
3. In a parallelogram  A B C D  prove that  (A C)^{2}+(B D)^{2}=2\left[(A B)^{2}+(B C)^{2}\right]

3. In a parallelogram A B C D prove that (A C)^{2}+(B D)^{2}=2\left[(A B)^{2}+(B C)^{2}\right]

Example: In a \triangle A B C with obtuse angle at A if \overline{C D} is an altitude on \overline{B A} produced and m \overline{A C}=m \overline{A B} Then prove that (B C)^{2}=2(A B)(B D)
Example: In  a \triangle A B C  with obtuse angle at  A  if  \overline{C D}  is an altitude on  \overline{B A}  produced and  m \overline{A C}=m \overline{A B} Then prove that  (B C)^{2}=2(A B)(B D)

Example: In a \triangle A B C with obtuse angle at A if \overline{C D} is an altitude on \overline{B A} produced and m \overline{A C}=m \overline{A B} Then prove that (B C)^{2}=2(A B)(B D)

2. In a \triangle A B C m \angle A=45^{\circ} prove that (B C)^{2}=(A B)^{2}+(A C)^{2}-\sqrt{2} m \overline{A B} m \overline{A C} .
2. In a  \triangle A B C m \angle A=45^{\circ}  prove that  (B C)^{2}=(A B)^{2}+(A C)^{2}-\sqrt{2} m \overline{A B} m \overline{A C} .

2. In a \triangle A B C m \angle A=45^{\circ} prove that (B C)^{2}=(A B)^{2}+(A C)^{2}-\sqrt{2} m \overline{A B} m \overline{A C} .

1. Given m \overline{A C}=1 \mathrm{~cm} m \overline{B C}=2 \mathrm{~cm} m \angle C=120^{\circ} .Compute the length A B and the area of \triangle A B C .Hint: (A B)^{2}=(A C)^{2}+(B C)^{2}+2 m A C . m C D where (m \overline{C D})=(m \overline{B C}) \cos \left(180^{\circ}-m \angle C\right) (Use theorem 1).
1. Given  m \overline{A C}=1 \mathrm{~cm} m \overline{B C}=2 \mathrm{~cm} m \angle C=120^{\circ} .Compute the length  A B  and the area of  \triangle A B C .Hint:  (A B)^{2}=(A C)^{2}+(B C)^{2}+2 m A C . m C D where  (m \overline{C D})=(m \overline{B C}) \cos \left(180^{\circ}-m \angle C\right) (Use theorem 1).

1. Given m \overline{A C}=1 \mathrm{~cm} m \overline{B C}=2 \mathrm{~cm} m \angle C=120^{\circ} .Compute the length A B and the area of \triangle A B C .Hint: (A B)^{2}=(A C)^{2}+(B C)^{2}+2 m A C . m C D where (m \overline{C D})=(m \overline{B C}) \cos \left(180^{\circ}-m \angle C\right) (Use theorem 1).

7. In a \triangle A B C a=17 \mathrm{~cm} b=15 \mathrm{~cm} and c=8 \mathrm{~cm} . Find m \angle A .
7. In a  \triangle A B C a=17 \mathrm{~cm} b=15 \mathrm{~cm}  and  c=8 \mathrm{~cm} . Find  m \angle A .

7. In a \triangle A B C a=17 \mathrm{~cm} b=15 \mathrm{~cm} and c=8 \mathrm{~cm} . Find m \angle A .

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