the hypotenuse of a right angled triangle is 25 cm amd its perimeter is 56 cm. find the length of the smallest side.
Answers
The area is 84 (cm)2.
Explanation:
Suppose that the lengths of the sides making the right angle are
xandy.
Since the hypotenuse is 25 cm, we have,
x2+y2=252=625....................(⋆1).
Further, the perimeter is 56 cm, so that, we get,
x+y+25=56,or,x+y=56−25=31.................(⋆2).
We get, from (⋆2),y=31−x.
Subst.ing in (⋆1),x2+(31−x)2=625.
∴x2+(312−62x+x2)−252=0.
∴2x2−62x+(31+25)(31−25)=0.
∴2x2−62x+56×6
Answer:
The length of the smallest side = 7cm
Step-by-step explanation:
Given,
The hypotenuse of a right-angled triangle = 25 cm
Perimeter = 56 cm.
To find,
The length of the smallest side
Recall the formulas
By the Pythagorean Theorem, the square of the hypotenuse of a right-angled triangle is equal to the sum of squares of the other two sides.
Perimeter of a triangle = sum of three sides of the triangle
(a+b)² = a²+b²+2ab
(a-b)² = a²+b² - 2ab
Solution:
Let a, b, and c be the length of the three sides of the right-angled triangle, and Let 'c' be the hypotenuse
Then given c = 25
Then by the Pythagorean Theorem, we have
c² = a² +b²
25² = a²+b²
a²+b² = 625 -------------------(1)
Also, Perimeter = a+b +c = 56
a+b +25 = 56
a+b = 56-25 = 31
a+b = 31 ------------(2)
Squaring equation(2) on both sides,
(a+b)² = 31² = 961
a²+b²+2ab = 961
Substitute the value of a²+b² from equation (1)
625+2ab = 961
2ab = 961-625 = 336
ab =168 -----------(3)
We know,
(a-b)² = a²+b² - 2ab
Substituting the value of a²+b² and ab in the above equation
(a-b)² = 625 - 2 ×168
= 289
a-b = 17 --------------(4)
Adding equation (2) and (4) we get
2a = 48
a = 24
from equation(2)
24+b = 31
b = 7
Hence the three sides of the triangle are 24cm,7cm,25cm
∴ The length of the smallest side = 7cm
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