Given that P - (1,1) E - (3,3) S-(7,3) and T - (9,1). IS PEST is an isosceles trapezoid? Justify your answer mathematically.

The true statements are:

TU ≅ TS

The length of line segment PR is 13 units.

Step-by-step explanation:

See the attached figure.

As shown, The circle is inscribed in triangle PRT.

There are 2 segments tangents to the circle from point P ⇒ PQ , PU

There are 2 segments tangents to the circle from point R ⇒ RQ , RS

There are 2 segments tangents to the circle from point T ⇒ TS , TU

The length of RS is 5, the length of PU is 8, and the length of UT is 6.

So, RS = RQ = 5   ,   PU = PQ = 8   ,   TU = TS = 6

We will check the options:

(1) The perimeter of the triangle is 19 units. ⇒ Wrong

Because: The perimeter of the triangle is 38 units.

The perimeter of the triangle = PR+RT+TP = (PQ+QR)+(RS+ST)+(TU+UP)=38

(2) TU ≅ TS ⇒ True

Because: TS , TU are tangents to the circle from point T

(3) PU ≅ TU ⇒ Wrong

Because: PU = 8  and TU = 6

(4) The length of line segment PR is 13 units. ⇒ True

Because: PR = PQ + QR = 8 + 5 = 13 units

(5) The length of line segment TR is 10 units. ⇒ Wrong

Because: TR = TS + SR = 6 + 5 = 11

The true statements are:

TU ≅ TS

The length of line segment PR is 13 units.