Answer:
Length of KS = 25 units and
Length of KP = 17 units.
Step-by-step explanation:
Given: KPST is a trapezoid, KP =ST, MN is a mid segment, h is the height =15
Also it is given that, MN = 20 , h = 15 and PS:KT = 3:7.
Since, the length of Mid-segment MN = 20.
Also, it is given: PS:KT = 3:7
Let PS = 3x and KT = 7x respectively.
then;
[tex]\frac{PS+KT}{2} = 20[/tex]
[tex]\frac{3x+7x}{2} = 20[/tex]
[tex]\frac{10x}{2} =20[/tex]
On Simplify, we get;
5x =20
Divide both sides by 5 we get;
x =4
Then:
Length of base PS = 3x = 3(4) = 12 and Length of base KT = 7x = 7(4) = 28.
Now, In triangle PLK
Using Pythagoras theorem to find KP;
It is given here, PL =h =15 and KL= 8 {you can see in the figure as shown below};
[tex]KP^2= PL^2 + KL^2[/tex]
[tex]KP^2 = 15^2+8^2 =225 +64 = 289[/tex]
[tex]KP = \sqrt{289}[/tex]
Simplify:
KP = 17
therefore, the length KP = 17 units
To, construct a line: Join K and S
Now, in triangle KRS
KR = KL +LR = 8 +12 =20 and SR = h= 15
Using Pythagoras theorem in KRS to find KS;
[tex]KS^2 = SR^2+KR^2[/tex]
[tex]KS^2 = 15^2 + 20^2 = 225 + 400 = 625[/tex]
[tex]KS = \sqrt{625}[/tex]
On simplify:
KS = 25
Therefore, the length of KS is, 25 units.
