Question

find the area of the triangle using the formula ( a=\frac{1}{2}bh ), and verify that the formula ( a=\frac{1}{2}absin c ) gives the same result.
the area, according to ( a=\frac{1}{2}bh ), is
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)
the area, using ( a=\frac{1}{2}absin c ), is
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Calculate area using \(A = \frac{1}{2}bh\)

Here, \(b = 2\) (base) and \(h = 4\) (height).

$$ A=\frac{1}{2}\times2\times4 $$

$$ A = 4 $$

Step2: Calculate area using \(A=\frac{1}{2}ab\sin C\)

Here, \(a = 4\sqrt{2}\), \(b = 2\), and \(C = 45^{\circ}\), \(\sin45^{\circ}=\frac{\sqrt{2}}{2}\)

$$ A=\frac{1}{2}\times4\sqrt{2}\times2\times\frac{\sqrt{2}}{2} $$

First, simplify \(\frac{1}{2}\times2 = 1\)

$$ A=4\sqrt{2}\times\frac{\sqrt{2}}{2} $$

Since \(\sqrt{2}\times\sqrt{2}=2\)

$$ A = 4\times\frac{2}{2} $$

$$ A = 4 $$

Answer:

The area, according to \(A=\frac{1}{2}bh\), is \(4\).
The area, using \(A = \frac{1}{2}ab\sin C\), is \(4\).