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Write True or False and justify your answer:

In a triangle, the sides are given as 11 cm, 12 cm and 13 cm. The length of the altitude is 10.25 cm corresponding to the side having length 12 cm.

Answers (1)

True

$Let \; a\; =\; 11\; cm ;\; b\; =\; 12\; cm\; ;\; c = 13 cm$

$Semi perimeter (S) = \frac{a+b+c}{2}$

$= \frac{11+12+13}{2}$

$= \frac{36}{2}=18cm$

$(Area\; of\; triangle \; ABC)\; by\; Heron's\; formula$

$=\sqrt{S\left ( S-a \right )\left ( S-b \right )\left ( S-c \right )}$

$=\sqrt{18\left ( 18-11 \right )\left ( 18-12 \right )\left ( 18-13\right )}$

$= \sqrt{18\times 7\times 6\times 5}$

$= \sqrt{6\times 3\times 7\times 6\times 5}$

$= \sqrt{6^{2}\times 3\times 7\times 5}$

$= 6\sqrt{3\times 7\times 5}$

$= 6\sqrt{105}$ $\because \sqrt{105}=10.2469\simeq 10.25$

$= 6 \times 10.25$

$= 61.5 cm^{2}$

$Given\; altitude\; =\; 10.25\; cm\; and$

$Its \; corresponding\; base \; = \; 12\; cm$

$Area\; of \times triangle ABC = \frac{1}{2} \times Base \times corresponding\; Height$

$= \frac{1}{2} \times 12 \times 10.25$

$=61.5cm^{2}$

Hence the area obtained is the same.

Therefore the given statement is true.

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