Since the triangle is isosceles, we know that 2 of the angles (that sum up to 180) must be equal. The question states that the noncongruent angles average 55°, thus providing us with a system of two equations:

Solving for x and y by substitution, we get x = 70° and y = 40° (which average out to 55°).

70 + 70 + 40 equals 180 also checks out.

Since 70° is not an answer choice for us, we know that the 40° must be one of the angles.

The measure of  is . Since , , and  are collinear, and the measure of  is , we know that the measure of  is .

Because the measures of the three angles in a triangle must add up to , and two of the angles in triangle  are  and , the third angle, , is .

Every triangle has 180 degrees.  An isosceles triangle has one vertex angle and two congruent base angles. 

Solve the equation for x to find the measure of the vertex angle. 

x = 180 - 27 - 27

x = 126

Therefore the measure of the vertex angle is .

Area of a triangle is ½ x base x height. Since base:height = 3:2, base = 1.5 height.  Area = 12 = ½ x 1.5 height x height or 24/1.5 = height².  Height = 4.  Base = 1.5 height = 6. Half the base and the height form the legs of a right triangle, with an equal leg of the isosceles triangle as the hypotenuse. This is a 3-4-5 right triangle.

This question is about the Triangle Inequality, which states that in a triangle with two sides A and B, the third side must be greater than the absolute value of the difference between A and B and smaller than the sum of A and B.

Applying the Triangle Inequality to this problem, we see that the third side must be greater than the absolute value of the difference between the other two sides, which is |6-6|=0, and smaller than the sum of the two other sides, which is 6+6=12. The only answer choice that does not satisfy this range of possible values is 12 since the third side must be LESS than 12.

This problem is solved using the geometric formula for the area of a triangle.  

Convert feet to inches.

Sketching ABC in the xy-plane, as pictured here, we see that it has base 6 and height 3. Since the formula for the area of a triangle is 1/2 * base * height, the area of ABC is 1/2 * 6 * 3 = 9.

When searching for the area of a triangle we are looking for the amount of the space enclosed by the triangle.

The equation for area of a triangle is

Plug the values for base and height into the equation yielding

Then multiply the numbers together to arrive at the answer .

If *, then the length of  must be .

Using the formula for the area of a triangle (), with , the area of the triangle must be .

We know that the perimeter is 36 inches, and one side is 12. This means, the sum of the lengths of the other two sides are 24. The ratio between the two sides is 3:5, giving a total of 8 parts. We divide the remaining length, 24 inches, by 8 giving us 3. This means each part is 3. We multiply this by the ratio and get 9:15, meaning the longest side is 15 inches.