Question

1. name the angle in three ways.
2. name the three different angles.
3. given: ( mangle abc = 40^circ )

find: ( mangle abd=)
( mangle dbe=)
( mangle abe=)

1. given:

( overrightarrow{ab} ) & ( overrightarrow{az} ) are opposite rays.
( mangle bal = 30^circ )
( mangle paz = 60^circ )
find: ( mangle lap=)
( mangle laz=)

1. find the distance between

( a(3,-4) ) & ( b(-2,5) )

1. find the midpoint of ( overline{ab} ).

( a(3,-4) )( b(-2,5) )

Explanation:

Step1: Apply the distance formula

The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). Here, \(x_1 = 3,y_1=-4,x_2=-2,y_2 = 5\).
Substitute into the formula: \(d=\sqrt{(-2 - 3)^2+(5-(-4))^2}\)

Step2: Simplify the expressions inside the square - root

First, calculate \((-2 - 3)=-5\) and \((5 + 4)=9\). Then \((-5)^2=25\) and \(9^2 = 81\).
So, \(d=\sqrt{25 + 81}\)

Step3: Calculate the sum and the square - root

\(25+81 = 106\). Then \(d=\sqrt{106}\approx10.3\)

Step4: Apply the mid - point formula

The mid - point formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(M=(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\).
Substitute \(x_1 = 3,y_1=-4,x_2=-2,y_2 = 5\) into the formula: \(M=(\frac{3+(-2)}{2},\frac{-4 + 5}{2})\)

Step5: Simplify the expressions for the mid - point coordinates

\(\frac{3-2}{2}=\frac{1}{2}\) and \(\frac{-4 + 5}{2}=\frac{1}{2}\)

Answer:

1. The distance between \(A(3,-4)\) and \(B(-2,5)\) is \(\sqrt{106}\approx10.3\)
2. The mid - point of \(\overline{AB}\) is \((\frac{1}{2},\frac{1}{2})\)