The observation tower has a height of 105 m above sea level. After D and B meet, two hours later, will D meet C. Trucks B and C start at place A while D starts at Y. Truck B is 60 km/h, C is 40 km/h and D is 80 km/h. There are places A and Y, three trucks, B, C, and D, respectively. The angle of elevation at the starting point is 42.4 degrees, and the angle of elevation at the end is 48.3 Determine the coefficient of sliding friction between the skis and the snow.Ī hiker plans to hike up one side of a mountain and down the other side of points a mountain, each side of the mountain formed by a straight line. It then moves to a horizontal section of the track, which travels the same length l until it stops. What is the change in distance between you and your friend from the beginning of the race?Ī skier starts down a hill of length l and an angle of inclination of 10˚. You are 3 2/3 feet ahead of your friend at the halfway point. How far can you see from the ship's mast, whose peak is at 14 meters above sea level? (Earth's radius is 6370 km).īefore a race, you start 4 5/8 feet behind your friend. Calculate how much meters altitude is higher upper station than at the base station. The horizontal distance between the upper and lower cable car station is 1600 m. Observation point A is 115m above the lake level. From the same place A, we see its image in the lake at a depth angle of 24° 12'. Determine the distance between the boats if both boats and the observer are in the same vertical plane.ĭetermine the height of the cloud above the lake's surface if we see it from place A at an elevation angle of 20° 57'. How wide is the river?Īn observer watches two boats at depth angles of 64° and 48° from the top of the hill, which is 75 m above the lake level. At 27 feet from the river's edge, the bridge is 9 feet above the water. The bridge is 10 feet above the water at the center of the river. The bridge over the river has the shape of an arc. What time do they meet, and how far will they be from point B? The distance between points A and B is 5.5km. Ondra will meet him from point B at a speed of 3 km/h at 9.30 am. Tomas will leave point A at a speed of 4 km/h at 9 o'clock. What is the horizontal distance of the airplane from the airport? It is flying at an altitude of 3 kilometers above the ground. The airplane sights a runway at an angle of depression of 23°. At what horizontal distance from point B should be release any body from the aircraft body to fall into point B? (g = 9.81 m/s²) The aircraft flies at an altitude of 12600 m above the ground at a speed of 532 km/h. The lowest place in this country, Nieuwerkerk, lies 7m below sea level. The highest point of the Netherlands, Vaalserberg, is only 328m above sea level. Determine the percentage decrease.Ī 5 km long road begins at an altitude 500 meters above sea level and ends at an altitude of 521 ASL. All triangles inscribed in a given circle and having the same orthocenter have the same nine-point circle.At a horizontal distance of 800 m, the ski slope drops to 220 m above sea level.A nine-point circle bisects a line segment going from the corresponding triangle’s orthocenter to any point on its circumcircle.The radius of a triangle’s circumcircle is twice the radius of the same triangle’s nine-point circle.Three Properties of the Nine-Point Triangle It was mathematician Olry Terquem who was the first to recognise the added significance of the three midpoints between the triangle’s vertices and the orthocenter. The nine-point circle is the complement to the circumcircle, which is the unique circle that passes through each of the triangle’s three vertices.Īlthough credited for its discovery, Karl Wilhelm Feuerbach did not entirely discover the nine-point circle, but rather the six point circle, as he only recognised that the midpoints of the three sides of the triangle and the feet of the altitudes of that triangle lay on the circle. On the contrary, for an obtuse triangle two of the altitudes have feet outside the triangle, but these feet still belong to the nine-point triangle. Note that for an acute triangle, six of the points – the midpoints and altitude feet – lie on the triangle itself.
Nine Point Circle | Source: Wolfram Mathworld
The midpoint of the line segment from each vertex of the triangle to point where the three altitudes meet, i.e.The foot of each altitude ( Ha, Hb, Hc).The midpoint of each side of the triangle ( Ma, Mb, Mc).It is named the nine-point circle as it passes through nine points defined from the triangle: The nine-point circle is a circle that can be that can be constructed for any given triangle.
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