Question

Consider pentagon PQRST. Starting at P and moving around the pentagon, the vertices are labelled P, Q, R, S, and T, in order. The pentagon has right angles at P, Q, and R, obtuse angles at S and T, and an area of 1000 cm2. Point V lies inside the pentagon such that ZPTV, ZTVS, and ZV SR are right angles. Point U lies on TV such that ASTU has an area of 210 cm². Also, it is known that PQ = 50 cm, SR= 15 cm, and TU = 30 cm. Determine the length of PT.​

Answer 1

The lengths of the opposite sides of a rectangle are equal

The length of $\overline{PT}$ is 15.8 cm

The reason why the above length is correct is as follows

$The \ area \ of \ a \ triangle = \dfrac{1}{2} \times Base \ length \times Height$

$The \ area \ of \ a \ triangle \ \Delta STU= \dfrac{1}{2} \times \overline {TU} \times \overline {SV}$

The area of triangle ΔSTU = 210 cm² (given)

$\overline{TU}$ = 30 cm (given)

Therefore;

$The \ area \ of \ a \ triangle \ \Delta STU= 210 \ cm^2 = \dfrac{1}{2} \times 30 \ cm} \times \overline {SV} = 15 \ cm} \times \overline {SV}$

$\overline {SV} = \dfrac{210 \ cm^2}{15 \ cm} = 14 \ cm$

Area of PQRST = Area of PWQR - Area of TWSZ

Area of PQRST + Area of TWSZ = Area of PWQR

Area of TWSZ = $\overline{TV}$ × $\overline{SV}$

In rectangle TWSV, $\overline{TV}$ = $\overline{SW}$

In rectangle PWQR, $\overline{PQ}$ = $\overline{WR}$  = $\overline{SW}$ + $\overline{SR}$ = 50

$\overline{SR}$ = 15 cm (given)

$\overline{SW}$ = 50 cm - 15 cm = 35 cm = $\overline{TV}$

Area of TWSZ = 35 cm × 14 cm = 490 cm²

Area of PWQR = 1,000 cm² + 490 cm² = 1,490 cm²

Area of PWQR = $\overline{PQ}$ × $\overline{PW}$

$\overline{PW} = \dfrac{Area \ of \ PWQR}{\overline{PQ}} = \dfrac{1,490 \ cm^2}{50 \ cm} = 29.8 \ cm$

$\overline{PW}$ = $\overline{PT}$ + $\overline{TW}$

$\overline{TW}$ = $\overline{SV}$

∴ $\overline{PW}$ = $\overline{PT}$ + $\overline{SV}$

$\overline{PT}$ =  $\overline{PW}$ - $\overline{SV}$

$\overline{PT}$ =  29.8 cm - 14 cm = 15.8 cm

$\overline{PT}$ = 15.8 cm

Learn more about the rectangle and triangle properties here:

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