How do you use Heron's formula to find the area of a triangle with sides of lengths $7$ $7$ , $4$ $4$ , and $9$ $9$ ?

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How do you use Heron's formula to find the area of a triangle with sides of lengths $7$ $7$ , $4$ $4$ , and $9$ $9$ ?

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Answer 1 · 2016-02-07T18:11:11

$A r e a = 13.416$ $Area=13.416$ square units

Explanation:

Heron's formula for finding area of the triangle is given by
$A r e a = \sqrt{s (s - a) (s - b) (s - c)}$ $Area=sqrt(s(s-a)(s-b)(s-c))$

Where $s$ $s$ is the semi perimeter and is defined as
$s = \frac{a + b + c}{2}$ $s=(a+b+c)/2$

and $a, b, c$ $a, b, c$ are the lengths of the three sides of the triangle.

Here let $a = 7, b = 4$ $a=7, b=4$ and $c = 9$ $c=9$

$\Rightarrow s = \frac{7 + 4 + 9}{2} = \frac{20}{2} = 10$ $implies s=(7+4+9)/2=20/2=10$

$\Rightarrow s = 10$ $implies s=10$

$\Rightarrow s - a = 10 - 7 = 3, s - b = 10 - 4 = 6 and s - c = 10 - 9 = 1$ $implies s-a=10-7=3, s-b=10-4=6 and s-c=10-9=1$
$\Rightarrow s - a = 3, s - b = 6 and s - c = 1$ $implies s-a=3, s-b=6 and s-c=1$

$\Rightarrow A r e a = \sqrt{10 \cdot 3 \cdot 6 \cdot 1} = \sqrt{180} = 13.416$ $implies Area=sqrt(10*3*6*1)=sqrt180=13.416$ square units

$\Rightarrow A r e a = 13.416$ $implies Area=13.416$ square units

Answer 2 · 2016-02-23T13:45:35

$13.416 . u n i t s$ $13.416. units$

Explanation:

Use Heron's formula:

Heron's formula:

$A r e a = \sqrt{s (s - a) (s - b) (s - c)}$ $color(blue)(Area=sqrt(s(s-a)(s-b)(s-c))$

Where,

$a - b - c = s i d e s, s = \frac{a + b + c}{2} = s e m i p e r i m e t e r$ $color(brown)(a-b-c=sides,s=(a+b+c)/2=semiperimeter$ $o f$ $color(brown)(of$ $△$ $color (brown)(triangle$

So,

$a = 7$ $color(red)(a=7$

$b = 4$ $color(red)(b=4$

$c = 9$ $color(red)(c=9$

$s = \frac{7 + 4 + 9}{2} = \frac{20}{2} = 10$ $color(red)(s=(7+4+9)/2=20/2=10$

Substitute the values

$\to A r e a = \sqrt{10 (10 - 7) (10 - 4) (10 - 9)}$ $rarrArea=sqrt(10(10-7)(10-4)(10-9))$

$\to = \sqrt{10 (3) (6) (1)}$ $rarr=sqrt(10(3)(6)(1))$

$\to = \sqrt{10 (18)}$ $rarr=sqrt(10(18))$

$\to = \sqrt{180}$ $rarr=sqrt180$

We can further simplify that,

$\sqrt{180} = \sqrt{36 \cdot 5} = 6 \sqrt{5} \approx 13.416 . u n i t s$ $color(green)(sqrt180=sqrt(36*5)=6sqrt5~~13.416.units$

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