the sides of an isosceles triangle are 8", 8" and 6". Find the length of the shortest altitude of the triangle

Answer 1

We first find the most obvious altitude (the altitude to the base 6). That is h = √(8² - 3²) = √55. (Pythagorean Theorem)

Thus, the area is
A = bh/2 = 6√55/2 = 3√55

Now, any combination of base and altitude will yield the same area, thus,
bh/2 = 3√55

We need to find the length of the altitude to the base 8, so
8h/2 = 3√55
4h = 3√55
h = 3/4 √55

Since we know that 3/4 √55 < √55, then the length of the shortest altitude of the triangle is 3/4 √55.

^_^

Answer 2

The longest altitude is 7.42", but the shortest altitude (there are two that are equal) is 5.56".

Here is a picture I drew for this problem:
http://img148.imageshack.us/img148/8112/trianglepn3.jpg
(I am not an artist, sorry for the crappy drawing!)

In this picture, triangle ADC is similar to triangle CEB because angle C = angle B, and angle ADC = angle CEB = 90°. Therefore, 3 / 8 = x / 6, because these sides MUST be proportional with similar triangles. Let us solve for x so we know the length of BE:
3 / 8 = x / 6 --- Cross-multiply...
8x = 18 ---- Divide by 8...
x = 2.25"

Now that we know two sides of the triangle CEB, we can solve for the other side (line CE) using the Pytharorean Theorom:
a² + b² = c²
(2.25)² + b² = (6)²
5.0625 + b² = 36 --- Subtract 5.0625 from both sides...
b² = 30.9375 --- Take the square root of each side...
b = 5.56" (approximately)

Answer 3

As the area of a triangle is given by 1/2Xbase Xaltitude,the altitude on the longer side of the triangle would give the shortest altitude.
Now.first of all find the area of the triangle,knowing all the three sides
Its half perimetr is 11
Therefore its area=(sqrt)(11X3X3X5)=22.25 inch^2
Now let a be the altitude on the 8" side
Therefore 1/2XaX8=22.25
=>a=22.25X2/8=5.5625 inches

Answer 4

Easy way: find the long altitide, H^2 = 8^2 + 3^3 = 73.

Area of the triangle is 1/2H*b = 1/2sqrt(73)*6 = 3sqrt(73)

Area of the triangle is also 1/2*8*(short altitude) = 4*h

So 4h = 3sqrt(73), h = 3/4sqrt(73)

Answer 5

It would be one of the two altitudes from the two vertices on the 6" side. And this would be sqrt(36 - 81/16) = sqrt(495)/4 = 3/4 * sqrt(55)

Answer 6

6=sqr(8x)
36=8x
x=4.5

a^2+b^2=c^2
(4.5)^2+b^2=6^2
20.25+b^2=36
b^2=15.75
b=3.97

the shortest is 3.97

Answer 7

I get 5.56" as well. Must be right :-)

Answer 8

7.4"