Question 927442
1.the sides of a triangle measures{{{a=20cm}}},{{{ c=25cm}}}, and {{{b=30cm}}}.


find the length of the angle bisector of the smallest angle.
find the length of the line segment joining the midpoints of the 20-cm and 30-cm sides.



Find the length of the altitude to the shortest side:

You know the lengths of 3 sides.
{{{a=20cm}}},{{{ c=25cm}}}, and {{{b=30cm}}}
h    is  the altitude

Use Heron's formula to find out the area

{{{A = sqrt( s ( s-a) ( s-b) ( s-c))}}} ........where{{{ s }}}= semi-perimeter and a,b, and c are side lengths

 {{{s = (20cm+25cm+30cm)/2=75cm/2=37.5cm}}}

{{{A = sqrt(37.5 ( 37.5-20) (37.5-25) (37.5-30))}}}

{{{A = sqrt(37.5 ( 17.5) (12.5) (7.5))}}}

{{{A = sqrt(61523.4375)}}}

{{{A = 248.04}}} then use

{{{248.04 = (1/2) * 20* h}}}

{{{248.04 = 10* h}}}

{{{h = 24.8 cm}}} ANSWER 

Find the length of the median to the longest side:

the longest side is {{{b=30 cm}}}

Draw triangle {{{ABC}}}, with
{{{AB = 25cm}}}, {{{BC = 20cm}}}, {{{AC = 30cm}}}

Using law of cosines, we get

{{{c^2 = a^2 + b^2-2*a*b*cos(C)}}}

{{{25^2 = 20^2 + 30^2-2*20*30*cos(C)}}}

{{{625= 400+ 900-1200*cos(C)}}}

{{{625= 1300-1200*cos(C)}}}

{{{1200*cos(C)= 1300-625}}}

{{{1200*cos(C)= 675}}}

{{{cos(C)= 675/1200}}}

{{{cos(C)= 0.5625}}}


Draw median from {{{B}}}  to {{{AC}}} at point {{{M}}}.

Now we have triangle {{{BCM}}} with {{{BC = 20cm}}}, {{{CM = 15cm}}}
We will find {{{BM}}} using law of cosines:

{{{BM^2= BC^2 + CM^2 - 2 (BC)( CM) cos(C)}}}

{{{BM^2= 20^2 + 15^2- 2 (20)(15) (0.5625)}}}

{{{BM^2= 400+ 225 - 337.5}}}


{{{BM^2= 287.5}}}

Median  {{{BM= sqrt(287.5)}}}=> {{{BM=16.96cm}}} ANSWER



find the length of the angle bisector of the smallest angle.

Then the length of the angle bisector {{{d}}} is given by:

    {{{d^2=(ac/(a+c)^2)((a+c)^2-b^2) }}}


{{{d^2=(20*25/(20+25)^2)((20+25)^2-30^2) }}}

{{{d^2=(500/(45)^2)((45)^2-30^2) }}}

{{{d^2=(500/(2025))(2025-900) }}}

{{{d^2=0.25(1125) }}}

{{{d^2=281.25}}}

{{{d=16.77}}} ANSWER


find the length of the line segment joining the midpoints of the 20-cm and 30-cm sides.

The mid-segment of a triangle (also called a midline) is a segment joining the midpoints of two sides of a triangle.

Properties:
1.
The mid-segment of a triangle joins the midpoints of two sides of a triangle such that it is parallel to the third side of the triangle.
	
2.
The mid-segment of a triangle joins the midpoints of two sides of a triangle such that its length is half the length of the third side of the triangle.

so, the third side is {{{25cm}}} and  the half of the third side is {{{12.5cm}}} ANSWER