Question

if p1,p2,p3 are altitudes of the triangle ABC,from the vertices and delta is the area of triangleABCthen 1/p1,1/p2,1/p3=?​

Answer 1

formula is half to base into height

Answer 2

$\frac{1}{P_{1}}+\frac{1}{P_{2}}+\frac{1}{P_{3}}=\frac{s}{\Delta }$

Step-by-step explanation:

1. Let area of triangle = Δ

2. Let

   $P_{1}$ is perpendicular to side AB.

   $P_{2}$ is perpendicular to side BC.

   $P_{3}$ is perpendicular to side CA.

3. Area of triangle $=\frac{1}{2}\times side\times altitude$

   $\Delta =\frac{1}{2}\times AB\times P_{1}$

   Which can also be written as

  $\frac{1}{P_{1}}=\frac{AB}{2\times \Delta }$   ...1)

4. Area of triangle $=\frac{1}{2}\times side\times altitude$

   $\Delta =\frac{1}{2}\times BC\times P_{2}$

   Which can also be written as

  $\frac{1}{P_{2}}=\frac{BC}{2\times \Delta }$   ...2)

5. Area of triangle $=\frac{1}{2}\times side\times altitude$

   $\Delta =\frac{1}{2}\times CA\times P_{3}$

   Which can also be written as

  $\frac{1}{P_{3}}=\frac{CA}{2\times \Delta }$   ...3)

6. Adding equation 1,2 and 3. We get

$\frac{1}{P_{1}}+\frac{1}{P_{2}}+\frac{1}{P_{3}}=\frac{AB}{2\times \Delta }+\frac{BC}{2\times \Delta }+\frac{CA}{2\times \Delta }$

On solving

$\frac{1}{P_{1}}+\frac{1}{P_{2}}+\frac{1}{P_{3}}=\frac{AB+BC+CA}{2\times \Delta }$      ...4)

7. Where Semi perimeter (s)$=\frac{AB+BC+CA}{2}$    ...5)

8. From equation 4 and 5

 $\frac{1}{P_{1}}+\frac{1}{P_{2}}+\frac{1}{P_{3}}=\frac{s}{\Delta }$