ABC is a triangle in which AB = AC = 4 cm and ZA = 90°. Calculate the area of ABC . Also find the length of perpendicular from A to BC.
Answers
Step-by-step explanation:
Area of triangle ABC = 1/2 ×BASE × HEIGHT
= 1/2 × AB × AC
= 1/2 × 4× 4
= 2 × 4
= 8 cm square
LET AD IS PERPENDICULAR TO BC
In Triangle ADB AND ADC
AD = AD ( common)
angle ADC = angle ADB ( each 90 degree)
AB = AC (Given)
Triangle ADB (congruent to). triangle ADC ( RHS criteria)
BD = CD ( cpct )
in triangle ABC , BY Pythagoras theorem
ABsquare + ACsquare = BC square
4 square +4 square = BCsquare
16 +16 = BCsquare
32 = BC square
:. BC = 4√2 cm
BD + CD = BC
2BD = BC (. BD = CD )
2BD = 4√2
BD = 2√2 cm
now in triangle BAD , by Pythagoras theorem
ADsquare + BD square = ABsquare
ADsquare + 2√2 square = 4 square
ADsquare + 4×2 = 16
ADsquare + 8 = 16
ADsquare = 16- 8
ADsquare = 8
.: AD = √8
.: AD = 2√2 cm
Hence length of perpendicular from A to BC = length of AD = 2√2 cm
Hope this helps
Given :-
AB = AC = 4 cm
Angle A = 90°
(In the Attachment)
Let ∆ABC be the given triangle and h be the length of altitude AD.
Then draw AD Perpendicular To BC.
Therefore, BD = ½BC
Now,
BC² = AB² + AC²
BC² = (4)² + (4)²
BC² = 16 + 16
BC = √32
BC = 5.65 cm
So,
BD = ½BC = ½ × 5.65 = 2.825 cm
Now,
AB² = AD² + BD²
(4)² = h² + (2.825)²
16 = h² + 7.980625
h² = 16 - 7.980625
h = √8.091675
h = 2.84 cm
Area a = ½ × b × h
= ½ × BC × AD
= ½ × 5.65 × 2.84
= 8.02 cm²
= 8 cm² [Estimation]
Length of perpendicular A to BC :
AD = 2.84 cm
