Chapter 8: Q. 48 (page 542)

If h₁ , h₂ , and h₃ are the altitudes dropped from P, Q, and R, respectively, in a triangle (see the figure shown to the right), show that $\frac{1}{h_{1}} + \frac{1}{h_{2}} + \frac{1}{h_{3}} = \frac{s}{K}$
Where K is the area of triangle and $s = \frac{1}{2} (a + b + c)$

Short Answer

Expert verified

We proved $\frac{1}{h_{1}} + \frac{1}{h_{2}} + \frac{1}{h_{3}} = \frac{s}{K}$

Step by step solution

Given information

We are given,

Find the area of triangle

The area of triangle can be given as

$K = \frac{a h_{1}}{2}$

Similarly it can also be given as

$K = \frac{b h_{2}}{2} A n d K = \frac{c h_{3}}{2}$

Now Write the areas in terms of a,b,c

We get,

$a = \frac{2 K}{h_{1}} b = \frac{2 K}{h_{2}} c = \frac{2 K}{h_{3}}$

Now substitute the values in the formula of s

We have

$s = \frac{1}{2} (a + b + c) s = \frac{1}{2} (\frac{2 K}{h_{1}} + \frac{2 K}{h_{2}} + \frac{2 K}{h_{3}}) s = \frac{K}{h_{1}} + \frac{K}{h_{2}} + \frac{K}{h_{3}} \frac{1}{h_{1}} + \frac{1}{h_{2}} + \frac{1}{h_{3}} = \frac{s}{K}$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

If h₁ , h₂ , and h₃ are the altitudes dropped from P, Q, and R, respectively, in a triangle (see the figure shown to the right), show that $\frac{1}{h_{1}} + \frac{1}{h_{2}} + \frac{1}{h_{3}} = \frac{s}{K}$
Where K is the area of triangle and $s = \frac{1}{2} (a + b + c)$

Short Answer

Step by step solution

Given information

Find the area of triangle

Now Write the areas in terms of a,b,c

Now substitute the values in the formula of s

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Pure Maths

Discrete Mathematics

Calculus

Probability and Statistics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

If h1 , h2 , and h3 are the altitudes dropped from P, Q, and R, respectively, in a triangle (see the figure shown to the right), show that1h1+1h2+1h3=sKWhere K is the area of triangle ands=12(a+b+c)

Short Answer

Step by step solution

Given information

Find the area of triangle

Now Write the areas in terms of a,b,c

Now substitute the values in the formula of s

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Pure Maths

Discrete Mathematics

Calculus

Probability and Statistics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

If h₁ , h₂ , and h₃ are the altitudes dropped from P, Q, and R, respectively, in a triangle (see the figure shown to the right), show that $\frac{1}{h_{1}} + \frac{1}{h_{2}} + \frac{1}{h_{3}} = \frac{s}{K}$
Where K is the area of triangle and $s = \frac{1}{2} (a + b + c)$