Analytic Geometry

In triangle ABC, angle \(C = 70^\circ\), \(A = 45^\circ\), \(AB = 40\) m. What is the length of the median drawn from vertex A to side BC?

From a point outside of an equilateral triangle, the distances to the vertices are \(10\) m, \(18\) m and \(10\) m, respectively. What is the length of one side of a triangle?

A square section ABCD has one of its sides equal to x. Point E is inside the square forming an equilateral triangle BEC having one side equal in length to the side of the square. Find the angle AED.

Two perpendicular chords both \(5\) cm from the center of a circle divide the circle into four parts. If the radius of the circle is \(13\) cm, find the area of the smallest part.

A swimming pool is constructed in the shape of two partially overlapping identical circles. Each of the circles has a radius of \(9\) m and each circle passes through the center of the other. Find the area of the swimming pool.

A rectangle ABCD which measures \(18\) cm. by \(24\) cm. is folded once, perpendicular to diagonal AC, so that the opposite vertices A and C coincide. Find the length of the fold.

Find the area of a quadrilateral having sides \(AB = 10\) cm, \(BC = 5\) cm. \(CD = 14.14\) cm and \(DA = 15\) cm, if the sum of the opposite angles is equal to \(225^\circ\).

Find the area of a quadrilateral have sides \(12\) m, \(20\) m, \(8\) m and \(16.97\) m. if the sum of the opposite angles is equal to \(225^\circ\) find the area of the quadrilateral.

In a circle of diameter of \(10\) m, a regular five-pointed star touching its circumference is inscribed. What is the area of that part not covered by the star?

A regular pentagon has sides of \(20\) cm. An inner pentagon with sides of \(10\) cm is inside and concentric to the larger pentagon. Determine the area inside and concentric to the larger pentagon but outside of the smaller pentagon.

A line passes through point \((2,2)\). Find the equation of the line if the length of the line segment intercepted by the coordinates axes is the square root of \(5\).

Determine the equation of the circle whose radius is 5, center on the line \(x=2\) and tangent to the line \(3x-4y+11=0\).

Find the equation of the hyperbola whose asymptotes are \(y=\pm2x\) and which passes through \((5/2,3)\).

Find the equation of the hyperbola with vertices at \((-4,2)\) and \((0,2)\) and foci at \((-5,2)\) and \((1,2)\).

The cable of a suspension bridge hangs in the form of a parabola when the load is uniformly distributed horizontally. The distance between two towers is 150 m, the points of the cable on the towers are 22 m above the roadway, and the lowest point on the cable is 7 m above the roadway. Find the vertical distance to the cable from a point in the roadway 15 m from the foot of a tower.

An equilateral triangle is inscribed in the parabola \(x^2 = 8y\) such that one of its vertices is at the origin. Find the length of the side of the triangle.