The length of the third side of a triangle must always be between (but not equal to) the sum and the difference of the other two sides. 

For instance, take the example of 2, 6, and 7.

 and . Therefore, the third side length must be greater than 4 and less than 8. Because 7 is greater than 4 and less than 8, it is possible for these to be the side lengths of a triangle.

The 5, 7, 12 answer choice is the only option for which this is not the case.

 and . Therefore, the third side length must be between 2 and 12. Because it is equal to the sum, not less than the sum, it is not possible that these could be the side lengths of a triangle.

The perimeter is equal to the sum of the three sides.  In similar triangles, each side is in proportion to its correlating side.  The perimeters are also in equal proportion.

Perimeter A = 45” and perimeter B = 135”

The proportion of Perimeter A to Perimeter B is . 

This applies to the sides of the triangle.  Therefore to get the any side of Triangle B, just multiply the correlating side by 3.

15” x 3 = 45”

10” x 3 = 30“

The area of a triangle = (1/2)bh where b is base and h is height. 18 = (1/2)4h which gives us 36 = 4h so h =9.

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.

If *, then the length of  must be .

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

A diagonal is a line segment joining two non-adjacent vertices of a polygon.  A regular hexagon has six sides and six vertices.  One vertex has three diagonals, so a hexagon would have three diagonals times six vertices, or 18 diagonals.  Divide this number by 2 to account for duplicate diagonals between two vertices. The formula for the number of vertices in a polygon is:

where .

A diagonal connects two non-consecutive vertices of a polygon.  A hexagon has six sides.  There are 3 diagonals from a single vertex, and there are 6 vertices on a hexagon, which suggests there would be 18 diagonals in a hexagon.  However, we must divide by two as half of the diagonals are common to the same vertices. Thus there are 9 unique diagonals in a hexagon. The formula for the number of diagonals of a polygon is:

where n = the number of sides in the polygon.

Thus a pentagon thas 5 diagonals.  An octagon has 20 diagonals.

The key is to examine  in thie figure below:

Each interior angle of a regular hexagon, including , measures , so it can be easily deduced by way of the Isosceles Triangle Theorem that . , so by angle addition, 

.

Also, by symmetry,

,

so ,

and  is a  triangle whose long leg  has length .

By the  Theorem, , which is the hypotenuse of , has length  times that of the long leg, so .

The key is to examine  in thie figure below:

Each interior angle of a regular hexagon, including , measures , so it can be easily deduced by way of the Isosceles Triangle Theorem that . , so by angle addition, 

.

Also, by symmetry,

,

so ,

and  is a  triangle whose hypotenuse  has length .

By the  Theorem, the long leg  of  has length  times that of hypotenuse , so .

The key is to examine  in thie figure below:

Each interior angle of a regular hexagon, including , measures , so it can be easily deduced by way of the Isosceles Triangle Theorem that . To find   we can subtract  from . Thus resulting in:

Also, by symmetry,

,

so .

Therefore,  is a  triangle whose short leg  has length  .

The hypotenuse  of this   triangle measures twice the length of short leg , so .