Question

S. Fie ABC cu AB = AC = 5 cm, BC = 6 cm şi AD perpendicular BC, D apartine (BC) 2p a) Determinati lungimea inaltimii AD. 3p b) Fie P apartine (AD) si PT perpendicular AC, T apartine (AC), Determinati lungimea segmentului PT, stiind ca PT = PD. ​

Answer 1

a)

AB=AC⇒ ΔABC isoscel ⇒ AD mediana⇒ D mijlocul lui BC⇒

BD=DC=3 cm

AB=AC=5 cm

BC=6 cm

In ΔADB dreptunghic aplicam Pitagora (suma catetelor la patrat este egala cu ipotenuza la patrat)

AB²=AD²+BD²

25=9+AD²

AD=4 cm

b)

ΔATP dreptunghic si ΔADC dreptunghic ⇒ T.F.A

$\frac{AT}{AD}=\frac{AP}{AC}=\frac{PT}{DC} \\\\\frac{AT}{4} =\frac{AP}{5}=\frac{PT}{3}$

$\frac{AP}{5}=\frac{PT}{3} \\\\AD=AP+PD\\\\AD=AP+PT\\\\4=AP+PT\\\\AP=4-PT$

$\frac{AP}{5}=\frac{PT}{3}$

3AP=5PT

Inlocuim AP si obtinem:

3(4-PT)=5PT

12-3PT=5PT

12=8PT

$PT=\frac{12}{8} =\frac{3}{2}$

PT=1,5 cm

Un alt exercitiu gasesti aici: https://brainly.ro/tema/3444260

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