The answer is \(\displaystyle 12\).

Since we know that the permieter is \(\displaystyle 40\) inches and one side is \(\displaystyle 12\) inches, it can be determined that the remaining two sides must combine to be \(\displaystyle 40-12=28\) inches. The ratio of the remaining two sides is \(\displaystyle 3:4\) which means 3 parts : 4 parts or 7 parts combined. We can then set up the equation \(\displaystyle 7x=28\), and divide both sides by \(\displaystyle 7\) which means \(\displaystyle x=4\). The ratio of the remaining side lengths then becomes \(\displaystyle 3*(4):4*(4)\) or \(\displaystyle 12:16\). We now know the 3 side lengths are \(\displaystyle 12,12,and\ 16\).

\(\displaystyle 12\) is the shortest side and thus the answer.

To form an isosceles triangle here, we need to create a third vertex whose \(\displaystyle y\) coordinate is between \(\displaystyle -1\) and \(\displaystyle 7\).  If a vertex is placed at \(\displaystyle (2,3)\), the distance from \(\displaystyle (1,-1)\) to this point will be \(\displaystyle \sqrt{17}\). The distance from \(\displaystyle (1,7)\) to this point will be the same.

Because \(\displaystyle \bigtriangleup BDE\) is isosceles, \(\displaystyle \measuredangle DBE\) equals \(\displaystyle \left(\frac{180-58}{2}\right)^{\circ}\) or \(\displaystyle 61^{\circ}\).

We know that \(\displaystyle \measuredangle ABE, \measuredangle DBE, \measuredangle CBD\) add up to \(\displaystyle 180^{\circ}\), so \(\displaystyle \measuredangle CBD\) must equal \(\displaystyle (180-90-61)^{\circ}\) or \(\displaystyle 29^{\circ}\).

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 \(\displaystyle 5\) feet apiece, which meets the requirement for isosceles triangles, and having one angle of \(\displaystyle 30^{\circ}\) at the vertex where the two congruent sides meet means the other two angles must be \(\displaystyle 75^{\circ}\) and \(\displaystyle 75^{\circ}\). The missing side connecting the two sensors, therefore, is opposite the \(\displaystyle 30^{\circ}\) 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 \(\displaystyle A, B\) and \(\displaystyle C,\) and opposite sides \(\displaystyle a, b\) and \(\displaystyle c\):

\(\displaystyle \frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}\).

Plugging in one of our \(\displaystyle 75^{\circ}\) angles (and its corresponding \(\displaystyle 5\) ft side) into this equation, as well as our \(\displaystyle 30^{\circ}\) angle (and its corresponding unknown side) into this equation gives us:

\(\displaystyle \frac{\sin 75^{\circ}}{5} = \frac{ \sin 30^{\circ}}{x}\)

Next, cross-multiply:

\(\displaystyle \frac{\sin 75^{\circ}}{5} = \frac{\sin 30^{\circ}}{x}\) ---> \(\displaystyle x\sin 75^{\circ} = 5\sin 30^{\circ}\)

Now simplify and solve:

\(\displaystyle x = \frac{5\sin 30^{\circ}}{\sin 75^{\circ}} = 2.588\)

Rounding, we see our missing side is \(\displaystyle 2.65\textup{ feet}\) long.

\(\displaystyle A=\frac{1}{2}bh\). 

\(\displaystyle 6x=42\)

\(\displaystyle x=7\)

The area of a triangle is given by the equation: 

\(\displaystyle \small A=\frac{1}{2}bh\)

In this case, we are given the area (\(\displaystyle \small A\)) and the base (\(\displaystyle \small b\)) and are asked to solve for height (\(\displaystyle \small h\)).

To do this, we must plug in the given values for \(\displaystyle \small A\) and \(\displaystyle \small b\), which gives the following:

\(\displaystyle \small 24\;cm^2=\frac{1}{2}(4\; cm)h\)

We then must multiply the right side, and then divide the entire equation by 2, in order to solve for \(\displaystyle \small h\):

\(\displaystyle \small 24\;cm^2=2\;cm(h)\)

\(\displaystyle \small \frac{24\; cm^2}{2\; cm}=\frac{2\; cm(h)}{2\; cm}\)

\(\displaystyle \small 12\;cm=h\)

\(\displaystyle \small h=12\;cm\)

Therefore, the height of the triangle is \(\displaystyle \small 12\;cm\).

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 \(\displaystyle 101^{\circ}\). That means the sum of the other two angles is \(\displaystyle 79^{\circ}\). The sum of the corresponding angles in triangle 2 is \(\displaystyle 80^{\circ}\). Therefore, because \(\displaystyle 79^{\circ}\) is not equal to \(\displaystyle 80^{\circ}\), the two obtuse triangles cannot be congruent.

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

\(\displaystyle 180 (n-2)\) degrees, where \(\displaystyle n\) 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: \(\displaystyle 180(4-2)\) degrees \(\displaystyle =180(2)\) degrees \(\displaystyle =360\) 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 \(\displaystyle 360\) degrees and the sum of the non-congruent opposite angles.

The solution is:

\(\displaystyle 40+25=65\)

\(\displaystyle 360-65=295\) degrees

Thus, \(\displaystyle 295\) degrees is the sum of the remaining two opposite angles.

Check:

\(\displaystyle 295+65=360\)

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

\(\displaystyle 180 (n-2)\) degrees, where \(\displaystyle n\) 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: \(\displaystyle 180(4-2)\) degrees \(\displaystyle =180(2)\) degrees \(\displaystyle =360\) 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 \(\displaystyle 360\) degrees and the non-congruent opposite angles sum by \(\displaystyle 2\):   

\(\displaystyle 28+84=112\)

\(\displaystyle 360-112=248\)

This means that \(\displaystyle 248\) is the sum of the remaining two angles, which must be opposite congruent angles. Therefore, the measurement for one of the angles is: 

\(\displaystyle \frac{248}{2}=124\) 

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

\(\displaystyle 180 (n-2)\) degrees, where \(\displaystyle n\) 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: \(\displaystyle 180(4-2)\) degrees \(\displaystyle =180(2)\) degrees \(\displaystyle =360\) 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 \(\displaystyle 360\) degrees and the non-congruent opposite angles sum by \(\displaystyle 2\):   

\(\displaystyle 118+51=169\)

\(\displaystyle 360-169=191\)

This means that \(\displaystyle 191\) is the sum of the remaining two angles, which must be opposite congruent angles. Therefore, the measurement for one of the angles is: 

\(\displaystyle \frac{191}{2}=95.5\)