The answer is .

Since we know that the permieter is  inches and one side is  inches, it can be determined that the remaining two sides must combine to be  inches. The ratio of the remaining two sides is  which means 3 parts : 4 parts or 7 parts combined. We can then set up the equation , and divide both sides by  which means . The ratio of the remaining side lengths then becomes  or . We now know the 3 side lengths are .

 is the shortest side and thus the answer.

To form an isosceles triangle here, we need to create a third vertex whose  coordinate is between  and .  If a vertex is placed at , the distance from  to this point will be . The distance from  to this point will be the same.

Because  is isosceles,  equals  or .

We know that  add up to , so  must equal  or .

This problem can be solved when one realizes that the light beam's split has resulted in an acute isosceles triangle. The triangle as stated has two sides of  feet apiece, which meets the requirement for isosceles triangles, and having one angle of  at the vertex where the two congruent sides meet means the other two angles must be  and . The missing side connecting the two sensors, therefore, is opposite the  angle.

Since we know at least two angles and at least one side of our triangle, we can use the Law of Sines to calculate the remainder. The Law of Sines says that for any triangle with angles  and  and opposite sides  and :

.

Plugging in one of our  angles (and its corresponding  ft side) into this equation, as well as our  angle (and its corresponding unknown side) into this equation gives us:

Next, cross-multiply:

 ---> 

Now simplify and solve:

Rounding, we see our missing side is  long.

. 

The area of a triangle is given by the equation: 

In this case, we are given the area () and the base () and are asked to solve for height ().

To do this, we must plug in the given values for  and , which gives the following:

We then must multiply the right side, and then divide the entire equation by 2, in order to solve for :

Therefore, the height of the triangle is .

In order for two obtuse triangles to be congruent, the sum of the two smaller angles must equal the sum of the two smaller angles of the second triangle. That is, excluding the obtuse angle. 

The first triangle has an obtuse angle of . That means the sum of the other two angles is . The sum of the corresponding angles in triangle 2 is . Therefore, because  is not equal to , the two obtuse triangles cannot be congruent.

The sum of the interior angles of any polygon can be found by applying the formula: 

 degrees, where  is the number of sides in the polygon. 

By definition, a kite is a polygon with four total sides (quadrilateral). The sum of the interior angles of any quadrilateral must equal:  degrees  degrees  degrees. Additionally, kites must have two sets of equivalent adjacent sides & one set of congruent opposite angles. 

To find the sum of the remaining two angles, determine the difference between  degrees and the sum of the non-congruent opposite angles.

The solution is:



 degrees

Thus,  degrees is the sum of the remaining two opposite angles.

Check:

The sum of the interior angles of any polygon can be found by applying the formula:

 degrees, where  is the number of sides in the polygon.

By definition, a kite is a polygon with four total sides (quadrilateral). The sum of the interior angles of any quadrilateral must equal:  degrees  degrees  degrees. Additionally, kites must have two sets of equivalent adjacent sides & one set of congruent opposite angles.

The missing angle can be found by finding the sum of the non-congruent opposite angles. Then divide the difference between  degrees and the non-congruent opposite angles sum by :   





This means that  is the sum of the remaining two angles, which must be opposite congruent angles. Therefore, the measurement for one of the angles is: 

 

The sum of the interior angles of any polygon can be found by applying the formula: 

 degrees, where  is the number of sides in the polygon. 

A kite is a polygon with four total sides (quadrilateral). The sum of the interior angles of any quadrilateral must equal:  degrees  degrees  degrees. Additionally, kites must have two sets of equivalent adjacent sides & one set of congruent opposite angles. 

The missing angle can be found by finding the sum of the non-congruent opposite angles. Then divide the difference between  degrees and the non-congruent opposite angles sum by :   





This means that  is the sum of the remaining two angles, which must be opposite congruent angles. Therefore, the measurement for one of the angles is: