Its all about altitude

Geometry Level 4

Two of the altitude of a scalene triangle is 4 and 12 units.if the remaining altitude is also a natural number,then maximum value is

8 3 5 2 7 4

1 solution

Ngọc Bùi Tiến
Jul 31, 2015

let $\Delta ABC$ which has the area of $S$ , the sides $a, b, c$ and the respective altitudes $h_{a}=4, h_{b}=12, h_{c}$ . According to triangle inequalities:

$a-b>c$

$\dfrac{2S}{h_{a}}-\dfrac{2S}{h_{b}}>\dfrac{2S}{h_{c}}$

$\dfrac{2S}{4}-\dfrac{2S}{12}>\dfrac{2S}{h_{c}}$

$\dfrac{1}{4}-\dfrac{1}{12}>\dfrac{1}{h_{c}}$

$\dfrac{1}{6}>\dfrac{1}{h_{c}}$

$h_{c}<6$

Hence $\boxed{h_{c}=5}$

is this a formula that side a=2S/h(a)

Harshi Singh - 5 years, 10 months ago

Since $h_{a}$ is the respective altitude of $a$ , $S=\dfrac{a\times h_{a}}{2}$

Ngọc Bùi Tiến - 5 years, 10 months ago

In this formula $S=\dfrac{a\times h_{a}}{2}$ IS $S$ EQUAL TO SEMI PERIMETER??

Harshi Singh - 5 years, 10 months ago

@Harshi Singh – $S$ is the area of triangle $ABC$ , as written above.

Ngọc Bùi Tiến - 5 years, 10 months ago

@Ngọc Bùi Tiến – oh ya!! it is written sorry

Harshi Singh - 5 years, 10 months ago

Did the same way. Since S is a factor on both the sides, I did not use it.

Niranjan Khanderia - 4 years, 9 months ago

Its all about altitude

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