The perimeter of an isosceles triangle is 40 and the length of the altitude to its base is 10. Find the length of a leg.

The length of a leg is 12.5 and its base is 15

Step-by-step explanation:

Using Heron's formula, the area of the triangle A = √[s(s - a)(s - b)(s - c)] where s = (a + b + c)/2. Now, for an isosceles, a = b and c = its base.

So,  A = √[s(s - a)(s - a)(s - c)] = √[s(s - a)²(s - c)] = (s - a)√[s(s - c)] and s = (a + a + c)/2 = a + c/2 ⇒ s = a + c/2 (1)

Given that the perimeter a + b + c = 2a + c = 40 = 2s ⇒ s = 40/2 = 20 and A = hc/2 where h = length of altitude to base = 10.

So, A = 10c/2 = 5c

So, 5c = (s - a)√[s(s - c)]

From (1) s - a = c/2.

So, 5c = (s - a)√[s(s - c)]

5c = (c/2)√[s(s - c)]

10 = √[s(s - c)]

squaring by sides, we have

100 = s(s - c) since s = 20,

s(s - c) = 100

20(20 - c) = 100

20 - c = 100/20

20 - c = 5

c = 20 - 5

c = 15

From (1),

s = a + c/2

a = s - c/2

= 20 - 15/2

= 20 - 7.5

= 12.5

Since a = b = 12.5

So, the length of a leg is 12.5 and its base is 15

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step-by-step explanation:

a 4/6 = 9/12   not proportional

step-by-step explanation:

to find which are not proportional, we can use cross products

if it is true, they are proportional.   if it is not true, they are not proportional.

a 4/6 = 9/12

4* 12 = 6*9

48 = 54

not this is not proportional  

b 2/3 = 12/18  

2* 18 = 3 * 12

36 = 36

this is proportional

c 5/4 = 10/8  

5*8 = 4*10

40 = 40

this is proportional

d 7/5 = 35/25

7*25 = 5 * 35

175 = 175

is proportional