the measure of one side of right angle triangle is 42 cm. if difference in length and it's hypotenuse and other side is 14 cm. find measure of 2 unknown sides
Answers
Answer:
Note : refer the figure in the attachment ]
let us suppose,
the triangle is triangle ABC
AB = 42m, hypotenuse = AC = ?
BC = ?
◾we have given the relation between, AC (hypotenuse) and BC ( other side)
i.e difference between lengths of hypotenuse and other side is 14m.
AC - BC = 14m
AC ( hypotenuse) = 14 + BC ....(1)
◾As we have given triangle is right angle triangle, /_ ABC = 90°
We know,
★Pythagoras theorem , ( hypotenuse) ^2 = ( side 1 )^2 + ( side 2 )^2
Therefor, by Pythagoras theorem,
(AC) ^2 = ( AB) ^2 + ( BC )^2
from equation (1) substitute the value of AC
(14 + BC)^2 = (42 ) ^2 + ( BC) ^2
⟹ ( (14 )^2 + 2 x (14 ) x ( BC) + (BC) ^2)= 1,764 - ( BC) ^2
⟹ 196+28(BC)+(BC)^2-(BC)^2 = 1,764
⟹ 28(BC) = 1764 - 196
⟹ ( BC) = 1,568 / 28
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◾Now, To find AC( hypotenuse) ,use Pythagoras theorem
(AC)^2 = ( AB)^2 + (BC)^2
( AC) ^2 = ( 42 )^2 + ( 56 )^2
= 1,764 + 3,136
= 4,900
AC = √ ( 4,900)
= 70 m
therefor,
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Now, we know the formula to find the area of triangle,
★Area of triangle = (1/2 )x base x height
here, base = side BC and height = side AB
Therefor, Area of triangle ABC
= (1/ 2 )x ( BC) x ( AB)
= ( 1/2 ) x ( 56 ) x ( 42 )
= 28 x 42
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Verification by heron's formula:
★heron's formula
Area of triangle
= √[ s (s - a) (s - b) (s - c) ]
first, we have to find s ( semiperimeter)
s =( a + b + c ) /2 [ since, a, b, c are the sides of triangle ]
S =[ ( AB) + (BC) +( AC) ] / 2
= ( 42 + 56 + 70 ) / 2
= ( 98 + 70 ) / 2
= 168 / 2
= 84
Therefor apply the heron's for area of triangle,
1,176
= √ [ s ( s - (AB)) ( s - (BC)) ( s - (AC))]
1,176 =√ [ 84 ( 84 - 42 ) ( 84 - 56 ) ( 84 - 70)]
1,176 = √[ 84 x ( 42 )x ( 28 ) x ( 14) ]
1,176 = √ [ 84 x 14 x 42 x 28 ]
1,176 = √ [ 1, 176 x 1, 176 ]
1,176 = 1, 176
hence verified
afnan has given in a very complex way .....
refer to his answer.............