Question

Given: FPST is a trapezoid,

FP=ST, m∠F=45°,
PF=8, PS=10
Find: The midsegment MN.

Answer 1

`4√(2)+10`

Explanation:

Given:

FPST is a trapezoid,

FP=ST, m∠F=45°,

PF=8, PS=10

To Find:

The Length of the mid-segment MN.

Solution:

This trapezoid FPST is isosceles, because FP=ST

Also

`m\angle F=m\angle T=45^(\circ).`

Draw the height PH. Triangle FPH is right triangle with two angles of measure 45°. This means that

FH = HP --------------(1)

By the Pythagorean theorem,

`FH^2+PH^2=PF^2,`

From (1)

`\\ \\2FH^2=8^2`

`\\ \\2FH^2=64`

`FH^2=(64)/(2)`

`\\FH^2=32`

`\\FH^2= 2* 2* 2 *2 * 2`

`\\FH = √(2* 2* 2 *2 * 2)`

`\\FH=4√(2).`

Since trapezoid FPST is isosceles, the base FT has the length

`FT=2FH+PS\\`

`FT =8√(2)+10.`

Then the length of the mid-segment is

`MN=(FT+PS)/(2)`

`MN=(8√(2)+10+10)/(2)`

`MN=(8√(2)+20)/(2)`

`MN=4√(2)+10.`