Geometry Answers

Pythagorean theorem Question

A hapless motorist is trying to find his friend's house, which is located 2.50 miles west of the intersection of State Street and First Avenue. He sets out from the intersection, and drives the same number of kilometers due south instead. How far in kilometers is he from his intended destination?

Two boats leave the same bank of sea at the same time. One goes 12km per hour in the direction N550E and other goes 16km per hour in the direction S650E. Find the distance between the boats at the end of two hours.

Q. IN THE GIVEN FIGURE WE HAVE X AND Y ARE THE MIDPOINTS OF AB AND BC RESPECTIVELY AND AX= CY. SHOW THAT AB=BC.

Find the magnitude of vector A = 3i - 2j + 2k
(a)1 (b)5 (c) 3 (d) 2

2.Find the focus, vertex, directrix, and axis of the parabola. Sketch the curve.
A.Y = x2- 2x + 3
B. X = y2 + 2x - 4
C.Y = -2x2 +4x + 5

3.Find the equations of the parabolas from the definition
A.Directrix X = 0, focus at (6,0)
B.Vertex (0,4), focus (0,2)
C.Vertex (-2,0), directrix X= 1
D.Focus (-1,-2) directrix: X - 2y + 3 = 0

ELLIPESE
1.Given the ellipse with equation 9x2 + 25y2 = 225, find the major and minor axes, the eccentricity, the coordinates of the foci and vertices. Sketch the ellipse.
2.Find the equation of the ellipse with eccentricity 3/4. Foci in the y-axis, center at origin, and passing through (6,4).

CIRCLES
1.Find the equation of the circle with center at (2,3), tangent to the line 3x +4y + 2 = 0. Answer x2 + y2 -4y -6y - 3 = 0.
2.Find the equation of the circle tangent to both axes, radius 6, in the second quadrant. Answer x2 + y2 + 12x - 12y +36 = 0.
3.Find the equation of the line tangent to the circle x2 + y2 + 2x -4y = 0 at P(1,3) on the circle.

PARABOLA
1.Find the equation of the parabola with vertex at the origin which satisfies the given additional condition:

A.Focus(-3,0)
B.Directrix: y-2
C.Passing through (2,3) and axis along the x-axis
D.Passing through (2,3) and axis along the y-axis

DIVISION OF LINE SEGMENT
1.The segment joining (1,3), and (4,6) is extended a distance equal to one-sixth of its own length. Find the terminal points.
2.The vertices of a triangle are (-3,-7), (3,1), and (-8,2). Find the intersection of its medians.
3.A circle has its center at (3,-2) and one end of a diameter at (7,2). Find the other end of the diameter.

DISTANCE FROM A POINT TO A LINE
1.Find the distance from the line 3x + 7y + 12 = 0 to (6,-7)
2.Find the distance from the line 2x - y + 4 = 0 to (2,8)
3.Find the distance of the line x + 4y - 7 = 0 to (-5,4)
4.Find the bisectors of the interior angles of the triangle whose sides are the lines 7x + y - 7 = 0; x + y + 1 = 0; and x + 7y -4 = 0.
5.Find the distance of the point (6,-3) from the line 2x - y + 4 = 0.
6.Find the bisector of the obtuse angle between the lines 11x + 2y - 7 = 0 and x + 2y = 0.

DISTANCE BETWEEN TWO POINTS
1.Show that the points (2,-3), (5,0), (2,3) and (-1,0), are the vertices of a square.
2.The distance (x,1) is 2√5 units from (2,3). Find x.
3.What are the coordinates of the point 3 units from the y-axis and at distance √5 from (5,3)?

EQUATIONS OF LINES
1.What is the equation of a line through (7,-3) and perpendicular to the line whose inclination is Arctan 2.
2.Show that lines 2x + 3y - 2 = 0, 3x - 2y + 23 = 0 and x - 5y + 12 = 0 are the sides of an isosceles triangle.
3.What is the equation of the line passing through (4, -7) and perpendicular to the line through (0, 1) and (3, -3).
4.Find the equations of the altitudes of the triangle with vertices at (1, -1), (5,2) and (-2,4). Where do they intersect?

SMNF is a regular triangular pyramid. SO (height)=6cm. Measure of the SEO angle is 60 degrees. Find: MF, apothem SE, total area of pyramid, volume of pyramid and the area of the SME triangle.

https://docs.google.com/drawings/d/1lB1C5bVx0ffa-QMdn5_jty58NssietYAFYhtdQDPLfY/edit?usp=sharing

For what value of x will the angle between the lines with direction ratios (x, 2, 4) and
(1, 0, 1) be o 45 ?