I. A(−5, −1); B(3, −5); C(5, 2) ar(△ABC) = ? II. A(5, 3); B(2, 5); C(−5, 3); D(−4, −3) ar(□ABCD) = ? shortest solution


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Question Number 204062 by BaliramKumar last updated on 05/Feb/24

$I . A (- 5, - 1); B (3, - 5); C (5, 2) a r (△ ABC) = ?$ $II . A (5, 3); B (2, 5); C (- 5, 3); D (- 4, - 3) a r (◻ ABCD) = ?$ $shortest solution$
	Answered by mr W last updated on 05/Feb/24

	$I .$ $A =∣ \frac{1}{2} \| \begin{matrix} 3 - (- 5) & - 5 - (- 1) \\ 5 - (- 5) & 2 - (- 1) \end{matrix} \| ∣$ $=∣ \frac{1}{2} \| \begin{matrix} 8 & - 4 \\ 10 & 3 \end{matrix} \| ∣$ $= \frac{24 + 40}{2} = 32$ $I I .$ $A =∣ \frac{1}{2} \| \begin{matrix} - 3 & 2 \\ - 10 & 0 \end{matrix} \| ∣ + ∣ \frac{1}{2} \| \begin{matrix} - 10 & 0 \\ - 9 & - 6 \end{matrix} \| ∣$ $= \frac{20}{2} + \frac{60}{2} = 40$
	Commented by BaliramKumar last updated on 05/Feb/24

	$any other solution$
	Commented by mr W last updated on 05/Feb/24

	$s u r e l y t h e r e a r e o t h e r m e t h o d s .$ $b u t t h i s i s t h e s h o r t e s t s o l u t i o n$ $i t h i n k .$
	Answered by deleteduser1 last updated on 05/Feb/24

	$A B = \sqrt{80}; B C = \sqrt{53}; A C = \sqrt{109}$ $[A B C] = \sqrt{\frac{(\sqrt{80} + \sqrt{53} + \sqrt{109}) (\sqrt{80} + \sqrt{53} - \sqrt{109) (\sqrt{80} + \sqrt{109} - \sqrt{53}) (\sqrt{109} + \sqrt{53} - \sqrt{80})}}{16}}$ $= \sqrt{\frac{(24 + 2 \sqrt{80 \times 53}) (- 24 + 2 \sqrt{53 \times 80})}{16}} = \sqrt{\frac{4 \times 53 \times 80 - 24^{2}}{16}}$ $= 32$
	Answered by deleteduser1 last updated on 05/Feb/24

	$T r a n s l a t e a n y v e r t e x s a y C (5, 2) \to C (0, 0)$ $\Rightarrow A (- 5, - 1) \to A (- 10, - 3) \land B (3, - 5) \to B (- 2, - 7)$ $\Rightarrow [A B C] =∣ \frac{1}{2} \| \begin{matrix} - 10 & - 2 \\ - 3 & - 7 \end{matrix} \| ∣= \frac{64}{2} = 32$
	Answered by deleteduser1 last updated on 05/Feb/24

	$L e t C (- 5, 3) \to (0, 0); [A B C D] = [A B D] + [B C D]$ $\Rightarrow A (5, 3) \to A (10, 0); B (2, 5) \to B (7, 2); D \to (1, - 6)$ $\Rightarrow [B C D] =∣ \frac{1}{2} \| \begin{matrix} 7 & 1 \\ 2 & - 6 \end{matrix} \| ∣= 22$ $L e t A \to (0, 0) \Rightarrow B \to (- 3, 2); D \to (- 9, - 6)$ $\Rightarrow [A B D] =∣ \frac{1}{2} \| \begin{matrix} - 3 & - 9 \\ 2 & - 6 \end{matrix} \| ∣= 18 \Rightarrow [A B C D] = 40$
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