Question

use the graph below to answer question 99

1. the triangle shown has vertices ( a(2,0), b(4,5) ),

and ( c(4,-2) ). determine its area.
a 5
b 6
c 7
d 8

1. what is the product of the complex numbers

( (3 - 5i) ) and ( (3 + 5i) )?
a ( -16 )
b 34
c ( 9 - 25i )
d ( 9 + 25i )

Explanation:

Step1: Find the base and height

The base of the triangle can be found by the difference in the \(x -\)coordinates of \(B\) and \(C\) (since they have the same \(x -\)coordinate for the vertical side). The \(x -\)coordinate of \(B\) and \(C\) is \(x = 4\). The height is the difference in the \(x -\)coordinates of \(A\) and \(B\) (or \(A\) and \(C\)).
The length of the vertical side (base) \(b=\vert5-(- 2)\vert=\vert5 + 2\vert=7\). The length of the horizontal side (height) \(h=\vert4 - 2\vert=2\).

Step2: Use the area formula for a triangle

The area formula for a triangle is \(A=\frac{1}{2}\times base\times height\).
Substitute \(b = 7\) and \(h=2\) into the formula: \(A=\frac{1}{2}\times7\times2\).

$$
LATEXBLOCK0
$$

for 100:

Step1: Use the formula \((a + bi)(a - bi)=a^{2}+b^{2}\)

For the complex numbers \((3 - 5i)\) and \((3 + 5i)\), where \(a = 3\) and \(b = 5\).
According to the formula \((a+bi)(a - bi)=a^{2}+b^{2}\), we substitute \(a = 3\) and \(b = 5\) into the formula.

$$
LATEXBLOCK0
$$

Since \(i^{2}=-1\), then \(9-25i^{2}=9-25\times(-1)\)

Step2: Simplify the expression

$$
LATEXBLOCK1
$$

Answer:

C. 7