Find the missing measure (x)

Question

The triangle is an isosceles triangle. The height is 6 with the altitude bisecting the base. The base is x while the sides of the triangle are unknown. The angles of the triangle on the base is both 60 degrees. I am supposed to use the Pythagorean theorem to find x the base of the triangle.

Answer 1

the altitude creates 2 right triangles, each with base half the base of the full triangle

each is area 6x/2 = 3x each triangle

It's actually an equilateral triangle, given 2 angles = 60 each. All 3 angles are 60 degrees. They sum to 180.

each 3rd side is the hypotenuse of a right triangle, h^2 = 6^2 + (h/2)^2

h^2 - h^2/4 = 36

(3/4)h^2 = 36

h^2 = (4/3)36 = 48

h = sqr48 = almost 7 = 4sqr3

half the base squared = (4sqr3)^2 - 36 = 16(3) -36 = 48-36 = 12

b/2^2 = 12

b/2 = sqr12 = 2sqr3

b = 4sqr3

Answer 2

Since the base angles are 60 degree each, all angles are 60 degree so it's an equilateral triangle. If base is x then other two sides are also x each. Applying Pythagorean theorem to the triangle with a side x, other side altitude of 36 am and third side of x/2 (as height bisects the base),

x^2 = (X/2)^2+ 6^2

x^2 = (x^2/4) + 36

(3/4)x^2 = 36

x^2 = 48

x = 4sqrt(3)

Check

(4sqrt(3))^2 = 48

(4sqrt(3))/2)^2 + 36 = 12 + 36 = 48

which is correct

Answer 3

If the base angles are both 60°, then the third angle is also 60° and the triangle is equilateral. That means that both sides and the base are equal to x. Half of the base then is x/2. The altitude splits the equilateral triangle into two right triangles with sides equal to 6 and x/2. The hypotenuse is x. By the Pythagorean Theorem:

6² + (x/2)² = x²

36 + x²/4 = x²

36 = 3x²/4

Multiply both sides by 4/3 and you get:

48 = x²

± √48 = x

± 4√3 = x

You can eliminate the negative solution because you can't have a negative side length.

Answer 4

Draw a picture of an isosceles triangle, then draw a vertical line from the apex down to the middle of the base. You should now have two right triangles. Looking at one of the two right triangles, since we know the angle in the middle of the base is 90º, and we're told the angle on the outside of the base is 60º, the angle at the top has no choice other than to be 30º. The 30-60-90 triangle is special, in that we know the lengths of the sides are always in the proportion [ 1 : 2 : "square root of 3" ] for [ a : c : b ]. Since the base of length x is bisected, we know the base of one of the two right triangles is a = x/2. Since we know the hypotenuse is always double the base, we know c = x. Finally, we were given b = 6. The pythagorean theorem states a-squared + b-squared = c-squared, so we know (x/2)-squared + 6-squared = x-squared. Solve for x.