In an isosceles triangle the base is a whole number and is 4 feet less than the sum of the two equal sides. the perimeter is a whole number between 0 and 75 feet. find the possible lengths of the equal sides

1 < x < 19

Step-by-step explanation:

The three sides of the isosceles triangle would be

x, x, and 2x-4

Since the perimeter is between 0 and 75 we can make an inequality

0 < x+x+(2x-4) < 75

0 < 4x-4 < 75

4< 4x < 79

1 < x < 79/4

1 < x < 19.75

(whole number)

1 < x <19

We have duplicated side length a and base b.  Since we have a triangle, we have the triangle inequalities:

a > 0

b > 0

2a > b

And a+b > a which doesn't add any information.

Since the perimeter p=2a+b is a whole number and b is a whole number we conclude p-b=2a is a whole number.  That doesn't mean a is a whole number, it could have a fractional part of 1/2.

The perimeter constraint is

0 < 2a + b ≤ 75

The relation between a and b is

b = 2a - 4

That automatically means 2a > b so we don't have to worry about that one.

0 < 2a + 2a - 4  ≤ 75

4 < 4a ≤ 79

1 < a  ≤ 79/4

1 < a  ≤ 19.75

So far that gives possible a values of 1.5, 2, 2.5, ..., 19, 19.5

But we still have our other constraints, in particular

b > 0

2a  - 4 > 0

2a > 4

a > 2

So our possible lengths of a are:

2.5, 3, 3.5, 4, 4.5, 5, 5.5, 6, 6.5, 7, 7.5, 8, 8.5, 9, 9.5, 10, 10.5, 11, 11.5, 12, 12.5, 13, 13.5, 14, 14.5, 15, 15.5, 16, 16.5, 17, 17.5, 18, 18.5, 19 and 19.5

Let's check a=2.5 and a=19.5

a=2.5, b= 2a-4 = 5-4 = 1, perimeter p=2.5+2.5+1=6,  seems good

a=19.5, b=2a-4=35, p=19.5+19.5+35=74, seems good

let's go outside the bounds,

a=2, b=2(2)-4=0, not a triangle, good

a=20, b=2(20)-4=36, p=20+20+36=76, over the limit, good

Seems right.

complete the steps to find the area of the trapezoid.

area of rectangle = 256   square units

area of triangle 1 = 24  square units

area of triangle 2 = 16  square units

area of triangle 3 =   96 square units

area of trapezoid = 120  square units

step-by-step explanation:

d) by the cross product property, ab^2 = bc multiplied by ad

step-by-step explanation:

we are given

in the given triangle abc

angle a is 90° and segment ad is perpendicular to segment bc

we can see that

triangle(abc) and triangle(abd) are similar because they have common side and same angle (90 degree)

so, triangles(abc) and abd are similar

since, they are similar

so, the ratio of their sides must be equal

we get

now, we can cross multiply

and we get

so, this could be step to prove it