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.
- If two sides of the trapezoid are equal then, it is an isosceles trapezoid.
- Mid-segment of a trapezoid is a line segment which connects the midpoints of the non-parallel sides.
- A trapezoid mid segment connects the midpoints of two congruent sides of the trapezoid and is parallel to the pair of parallel sides.
- Length of the mid segment is the sum of two bases divide by 2
Since, the length of Mid-segment MN = 20.
Also, it is given: PS:KT = 3:7
Let PS = 3x and KT = 7x respectively.
then;
![\frac{PS+KT}{2} = 20](https://tex.z-dn.net/?f=%5Cfrac%7BPS%2BKT%7D%7B2%7D%20%3D%2020)
![\frac{3x+7x}{2} = 20](https://tex.z-dn.net/?f=%5Cfrac%7B3x%2B7x%7D%7B2%7D%20%3D%2020)
![\frac{10x}{2} =20](https://tex.z-dn.net/?f=%5Cfrac%7B10x%7D%7B2%7D%20%3D20)
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};
![KP^2= PL^2 + KL^2](https://tex.z-dn.net/?f=KP%5E2%3D%20PL%5E2%20%2B%20KL%5E2)
![KP^2 = 15^2+8^2 =225 +64 = 289](https://tex.z-dn.net/?f=KP%5E2%20%3D%2015%5E2%2B8%5E2%20%3D225%20%2B64%20%3D%20289)
![KP = \sqrt{289}](https://tex.z-dn.net/?f=KP%20%3D%20%5Csqrt%7B289%7D)
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;
![KS^2 = SR^2+KR^2](https://tex.z-dn.net/?f=KS%5E2%20%3D%20SR%5E2%2BKR%5E2)
![KS^2 = 15^2 + 20^2 = 225 + 400 = 625](https://tex.z-dn.net/?f=KS%5E2%20%3D%2015%5E2%20%2B%2020%5E2%20%3D%20225%20%2B%20400%20%3D%20625)
![KS = \sqrt{625}](https://tex.z-dn.net/?f=KS%20%3D%20%5Csqrt%7B625%7D)
On simplify:
KS = 25
Therefore, the length of KS is, 25 units.