Answer:
Length of KS and KP are; 25 and 17
Step-by-step explanation:
Given: KPST is a trapezoid, KP =ST, MN is a mid segment, h is the height =15
Also, 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 bases 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]
Simplify:
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.
In triangle PLK
Using Pythagoras theorem to find KP;
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]
or
[tex]KP = \sqrt{289} = 17[/tex]
therefore, the length KP = 17
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]
or
[tex]KS = \sqrt{625} = 25[/tex]
Therefore, the length of KS is, 25
