The perimeter of a triangle is the sum of the lengths of its sides:

p = x + y + z
p = 12cm + 11cm + 6cm = 29cm

A circle is a figure in which each point around its perimeter is an equal distance from the center. The radius of a circle is the distance between the center and any point along its perimeter. A chord is a line segment that connects any two points along its perimeter. The diameter of a circle is the length of a chord that passes through the center of the circle and equals twice the circle's radius (2r).

An exterior angle of a triangle is equal to the sum of the two interior angles that are opposite:

d° = b° + c°

To find angle c, remember that the sum of the interior angles of a triangle is 180°:

180° = a° + b° + c°
c° = 180° - a° - b°
c° = 180° - 46° - 41° = 93°

So, d° = 41° + 93° = 134°

A shortcut to get this answer is to remember that angles around a line add up to 180°:

a° + d° = 180°
d° = 180° - a°
d° = 180° - 46° = 134°

The Pythagorean theorem defines the relationship between the side lengths of a right triangle. The length of the hypotenuse squared (c²) is equal to the sum of the two perpendicular sides squared (a² + b²): c² = a² + b² or, solved for c, \(c = \sqrt{a + b}\)

You need to find the value of c so solve the first equation in terms of x:

-5c + x = 7
x = 7 + 5c

then substitute the result (7 - -5c) into the second equation:

7c + 7(7 + 5c) = -6
7c + (7 x 7) + (7 x 5c) = -6
7c + 49 + 35c = -6
7c + 35c = -6 - 49
42c = -55
c = \( \frac{-55}{42} \)
c = -1\(\frac{13}{42}\)