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Question Number 10006 by RasheedSoomro last updated on 20/Jan/17

A triangle is divided into four triangles and three quadrilaterals by three straight line segments(for figure see comment below). ^• The sum of the perimeters of the three quadri- laterals is 25 cm. ^• The sum of the perimeters of the four triangles is 20 cm. ^• The perimeter of the whole triangle is 19 cm. ^• What is the sum of the lengths of the three straight line segments?

$$\mathrm{A}\:\mathrm{triangle}\:\mathrm{is}\:\mathrm{divided}\:\mathrm{into}\:\mathrm{four}\:\mathrm{triangles} \\ $$$$\mathrm{and}\:\mathrm{three}\:\mathrm{quadrilaterals}\:\mathrm{by}\:\mathrm{three}\:\mathrm{straight} \\ $$$$\mathrm{line}\:\mathrm{segments}\left(\mathrm{for}\:\mathrm{figure}\:\mathrm{see}\:\mathrm{comment}\:\mathrm{below}\right). \\ $$$$ \\ $$$$\:^{\bullet} \mathrm{The}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{perimeters}\:\mathrm{of}\:\mathrm{the}\:\mathrm{three}\:\mathrm{quadri}- \\ $$$$\mathrm{laterals}\:\mathrm{is}\:\mathrm{25}\:\mathrm{cm}. \\ $$$$\:^{\bullet} \mathrm{The}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{perimeters}\:\mathrm{of}\:\mathrm{the}\:\mathrm{four}\:\mathrm{triangles} \\ $$$$\mathrm{is}\:\mathrm{20}\:\mathrm{cm}. \\ $$$$\:^{\bullet} \mathrm{The}\:\mathrm{perimeter}\:\mathrm{of}\:\mathrm{the}\:\mathrm{whole}\:\mathrm{triangle}\:\:\mathrm{is}\:\mathrm{19}\:\mathrm{cm}. \\ $$$$\:^{\bullet} \mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{lengths}\:\mathrm{of}\:\mathrm{the}\:\mathrm{three} \\ $$$$\mathrm{straight}\:\mathrm{line}\:\mathrm{segments}? \\ $$

Commented by RasheedSoomro last updated on 20/Jan/17

Answered by mrW1 last updated on 21/Jan/17

Q=(sum of the perimeters of the three quadrilaterals) T=(sum of the perimeters of the four triangles) P=(perimeter of the whole triangle) S=(sum of the lengths of the three straight line segments) Q+T=P+2S S=((Q+T−P)/2)=((25+20−19)/2)=13 cm

$$\mathrm{Q}=\left(\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{perimeters}\:\mathrm{of}\:\mathrm{the}\:\mathrm{three}\:\mathrm{quadrilaterals}\right) \\ $$$${T}=\left(\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{perimeters}\:\mathrm{of}\:\mathrm{the}\:\mathrm{four}\:\mathrm{triangles}\right) \\ $$$${P}=\left(\mathrm{perimeter}\:\mathrm{of}\:\mathrm{the}\:\mathrm{whole}\:\mathrm{triangle}\right) \\ $$$${S}=\left(\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{lengths}\:\mathrm{of}\:\mathrm{the}\:\mathrm{three}\:\mathrm{straight}\:\mathrm{line}\:\mathrm{segments}\right) \\ $$$${Q}+{T}={P}+\mathrm{2}{S} \\ $$$${S}=\frac{{Q}+{T}−{P}}{\mathrm{2}}=\frac{\mathrm{25}+\mathrm{20}−\mathrm{19}}{\mathrm{2}}=\mathrm{13}\:{cm} \\ $$

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