Question

The sides of a triangle are 40 cm, 41 cm and 9 cm. The length of its smallest altitude
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Answer 1

Step-by-step explanation:

the sides make a right angle triangle

so hypotenuse = 41

therefore it's altitude lies on hypotenuse

and it's length is half of hypotenuse

so length of altitude =41/2=20.5

Answer 2

The length of its smallest altitude is 8.78 cm

Step-by-step explanation:

sides of triangle a,b and c are 40 , 41 and 9 cm

now using herons formula

$s= \frac{a+b+c}{2} \\s= \frac{40+41+9}{2}\\ s= \frac{90}{2}\\ s= 45\\now,\\area = \sqrt{s(s-a)(s-b)(s-c)} \\area = \sqrt{45(45-40)(45-41)(45-9)}\\ area = \sqrt{45\times 5\times4\times36}\\ area = \sqrt{5\times3\times3\times5\times2\times2\times6\times6} \\area = 5\times3\times2\times6\\area = 180 cm^2$

as the hypotenuse is the longest side , in any right angles triangle , then smallest altitude is on the hypotenuse.

therefore

base = 41 cm

area = 180

area of triangle = $\frac{1}{2}\times base \times height$

$180 = \frac{1}{2} \times 41\times h\\\\\frac{180\times 2}{41} = h\\ h = 8.78 cm$

hence ,

The length of its smallest altitude is 8.78 cm

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