Misplaced Pages

#82917

42-452: Rather than a single straight barrel, the SHARP gun uses an L-shape design with two separate sections; the 270 ft (82 m) long steel combustion section & pump tube section is connected to the 155 ft (47 m) long launch tube (or barrel) at a right angle . 100,000 kg (220,000 lb) rail-mounted sleds sit at both ends of the pump tube to absorb recoil energy from firing and

84-448: A , m b , m c {\displaystyle m_{a},m_{b},m_{c}} is a right triangle if and only if any one of the statements in the following six categories is true. Each of them is thus also a property of any right triangle. The trigonometric functions for acute angles can be defined as ratios of the sides of a right triangle. For a given angle, a right triangle may be constructed with this angle, and

126-469: A , b , c {\displaystyle a,b,c} satisfying this equation. This theorem was proven in antiquity, and is proposition I.47 in Euclid's Elements : "In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle." As with any triangle, the area is equal to one half the base multiplied by the corresponding height. In

168-404: A rectangle which has been divided along its diagonal . When the rectangle is a square , its right-triangular half is isosceles , with two congruent sides and two congruent angles. When the rectangle is not a square, its right-triangular half is scalene . Every triangle whose base is the diameter of a circle and whose apex lies on the circle is a right triangle, with the right angle at

210-400: A right angle is an angle of exactly 90 degrees or ⁠ π {\displaystyle \pi } / 2 ⁠ radians corresponding to a quarter turn . If a ray is placed so that its endpoint is on a line and the adjacent angles are equal, then they are right angles. The term is a calque of Latin angulus rectus ; here rectus means "upright", referring to

252-565: A semicircle (with a vertex on the semicircle and its defining rays going through the endpoints of the semicircle) is a right angle. Two application examples in which the right angle and the Thales' theorem are included (see animations). The solid angle subtended by an octant of a sphere (the spherical triangle with three right angles) equals π /2  sr . Right triangle A right triangle or right-angled triangle , sometimes called an orthogonal triangle or rectangular triangle ,

294-527: A proof as well but using a more explicit assumption. In Hilbert 's axiomatization of geometry this statement is given as a theorem, but only after much groundwork. One may argue that, even if postulate 4 can be proven from the preceding ones, in the order that Euclid presents his material it is necessary to include it since without it postulate 5, which uses the right angle as a unit of measure, makes no sense. A right angle may be expressed in different units: Throughout history, carpenters and masons have known

336-535: A quick way to confirm if an angle is a true right angle. It is based on the Pythagorean triple (3, 4, 5) and the rule of 3-4-5. From the angle in question, running a straight line along one side exactly three units in length, and along the second side exactly four units in length, will create a hypotenuse (the longer line opposite the right angle that connects the two measured endpoints) of exactly five units in length. Thales' theorem states that an angle inscribed in

378-465: A right angle) and obtuse angles (those greater than a right angle). Two angles are called complementary if their sum is a right angle. Book 1 Postulate 4 states that all right angles are equal, which allows Euclid to use a right angle as a unit to measure other angles with. Euclid's commentator Proclus gave a proof of this postulate using the previous postulates, but it may be argued that this proof makes use of some hidden assumptions. Saccheri gave

420-565: A right angle. Right angles are fundamental in Euclid's Elements . They are defined in Book 1, definition 10, which also defines perpendicular lines. Definition 10 does not use numerical degree measurements but rather touches at the very heart of what a right angle is, namely two straight lines intersecting to form two equal and adjacent angles. The straight lines which form right angles are called perpendicular. Euclid uses right angles in definitions 11 and 12 to define acute angles (those smaller than

462-426: A right triangle are related by the Pythagorean theorem , which in modern algebraic notation can be written where c {\displaystyle c} is the length of the hypotenuse (side opposite the right angle), and a {\displaystyle a} and b {\displaystyle b} are the lengths of the legs (remaining two sides). Pythagorean triples are integer values of

SECTION 10

#1732884058083

504-528: A right triangle, if one leg is taken as the base then the other is height, so the area of a right triangle is one half the product of the two legs. As a formula the area T {\displaystyle T} is where a {\displaystyle a} and b {\displaystyle b} are the legs of the triangle. If the incircle is tangent to the hypotenuse A B {\displaystyle AB} at point P , {\displaystyle P,} then letting

546-430: A smaller 10,000 kg (22,000 lb) sled is mounted on a perpendicular set of tracks at the aft-end of the launch-tube near the junction point. The firing sequence begins with the ignition of a methane gas mixture in the combustion section behind the piston at the far end of the pump tube. The resultant explosion rapidly drives the 1,000 kg (2,200 lb) steel piston down the pump tube and further compresses

588-560: A triangle is a right triangle . In Unicode , the symbol for a right angle is U+221F ∟ RIGHT ANGLE ( ∟ ). It should not be confused with the similarly shaped symbol U+231E ⌞ BOTTOM LEFT CORNER ( ⌞, ⌞ ). Related symbols are U+22BE ⊾ RIGHT ANGLE WITH ARC ( ⊾ ), U+299C ⦜ RIGHT ANGLE VARIANT WITH SQUARE ( ⦜ ), and U+299D ⦝ MEASURED RIGHT ANGLE WITH DOT ( ⦝ ). In diagrams,

630-427: Is a triangle in which two sides are perpendicular , forming a right angle ( 1 ⁄ 4 turn or 90 degrees ). The side opposite to the right angle is called the hypotenuse (side c {\displaystyle c} in the figure). The sides adjacent to the right angle are called legs (or catheti , singular: cathetus ). Side a {\displaystyle a} may be identified as

672-427: Is a right triangle with a right angle at A . {\displaystyle A.} The converse states that the hypotenuse of a right triangle is the diameter of its circumcircle . As a corollary, the circumcircle has its center at the midpoint of the diameter, so the median through the right-angled vertex is a radius, and the circumradius is half the length of the hypotenuse. The following formulas hold for

714-509: Is destroyed and the hydrogen drives the projectile down a 4 in (100 mm) diameter barrel at extremely high velocities until it bursts through a thin plastic sheet covering the end of the gun. All recoil forces are absorbed by the rail-mounted sleds as they are propelled outwards along their tracks. Headed by John Hunter , the SHARP gun fired projectiles using expanding hydrogen and achieved velocities of 3 km/s (6,700 mph) or Mach 8.8 for 5 kg (11 lb) projectiles. Had

756-434: Is given by This formula only applies to right triangles. If an altitude is drawn from the vertex with the right angle to the hypotenuse then the triangle is divided into two smaller triangles which are both similar to the original and therefore similar to each other. From this: In equations, where a , b , c , d , e , f {\displaystyle a,b,c,d,e,f} are as shown in

798-449: Is the golden ratio . Since the sides of this right triangle are in geometric progression , this is the Kepler triangle . Thales' theorem states that if B C {\displaystyle BC} is the diameter of a circle and A {\displaystyle A} is any other point on the circle, then △ A B C {\displaystyle \triangle ABC}

840-408: Is the only triangle having two, rather than one or three, distinct inscribed squares. Given any two positive numbers h {\displaystyle h} and k {\displaystyle k} with h > k . {\displaystyle h>k.} Let h {\displaystyle h} and k {\displaystyle k} be the sides of

882-621: The geometric mean , and the arithmetic mean of two positive numbers a {\displaystyle a} and b {\displaystyle b} with a > b . {\displaystyle a>b.} If a right triangle has legs H {\displaystyle H} and G {\displaystyle G} and hypotenuse A , {\displaystyle A,} then where ϕ = 1 2 ( 1 + 5 ) {\displaystyle \phi ={\tfrac {1}{2}}{\bigl (}1+{\sqrt {5}}{\bigr )}}

SECTION 20

#1732884058083

924-451: The medians of a right triangle: The median on the hypotenuse of a right triangle divides the triangle into two isosceles triangles, because the median equals one-half the hypotenuse. The medians m a {\displaystyle m_{a}} and m b {\displaystyle m_{b}} from the legs satisfy In a right triangle, the Euler line contains

966-400: The semi-perimeter be s = 1 2 ( a + b + c ) , {\displaystyle s={\tfrac {1}{2}}(a+b+c),} we have | P A | = s − a {\displaystyle |PA|=s-a} and | P B | = s − b , {\displaystyle |PB|=s-b,} and the area

1008-460: The apex and the hypotenuse as the base; conversely, the circumcircle of any right triangle has the hypotenuse as its diameter. This is Thales' theorem . The legs and hypotenuse of a right triangle satisfy the Pythagorean theorem : the sum of the areas of the squares on two legs is the area of the square on the hypotenuse, a 2 + b 2 = c 2 . {\displaystyle a^{2}+b^{2}=c^{2}.} If

1050-466: The diagram. Thus Moreover, the altitude to the hypotenuse is related to the legs of the right triangle by For solutions of this equation in integer values of a , b , c , f , {\displaystyle a,b,c,f,} see here . The altitude from either leg coincides with the other leg. Since these intersect at the right-angled vertex, the right triangle's orthocenter —the intersection of its three altitudes—coincides with

1092-414: The diameter of the incircle is less than half the hypotenuse, and more strongly it is less than or equal to the hypotenuse times ( 2 − 1 ) . {\displaystyle ({\sqrt {2}}-1).} In a right triangle with legs a , b {\displaystyle a,b} and hypotenuse c , {\displaystyle c,} with equality only in

1134-468: The fact that an angle is a right angle is usually expressed by adding a small right angle that forms a square with the angle in the diagram, as seen in the diagram of a right triangle (in British English, a right-angled triangle) to the right. The symbol for a measured angle, an arc, with a dot, is used in some European countries, including German-speaking countries and Poland, as an alternative symbol for

1176-543: The isosceles case. If the altitude from the hypotenuse is denoted h c , {\displaystyle h_{c},} then with equality only in the isosceles case. If segments of lengths p {\displaystyle p} and q {\displaystyle q} emanating from vertex C {\displaystyle C} trisect the hypotenuse into segments of length 1 3 c , {\displaystyle {\tfrac {1}{3}}c,} then The right triangle

1218-412: The isosceles right triangle or 45-45-90 triangle which can be used to evaluate the trigonometric functions for any multiple of 1 4 π . {\displaystyle {\tfrac {1}{4}}\pi .} Let H , {\displaystyle H,} G , {\displaystyle G,} and A {\displaystyle A} be the harmonic mean ,

1260-552: The legs can be expressed in terms of the inradius and the other leg as A triangle △ A B C {\displaystyle \triangle ABC} with sides a ≤ b < c {\displaystyle a\leq b<c} , semiperimeter s = 1 2 ( a + b + c ) {\textstyle s={\tfrac {1}{2}}(a+b+c)} , area T , {\displaystyle T,} altitude h c {\displaystyle h_{c}} opposite

1302-402: The lengths of all three sides of a right triangle are integers, the triangle is called a Pythagorean triangle and its side lengths are collectively known as a Pythagorean triple . The relations between the sides and angles of a right triangle provides one way of defining and understanding trigonometry , the study of the metrical relationships between lengths and angles. The three sides of

Super High Altitude Research Project - Misplaced Pages Continue

1344-402: The longest side, circumradius R , {\displaystyle R,} inradius r , {\displaystyle r,} exradii r a , r b , r c {\displaystyle r_{a},r_{b},r_{c}} tangent to a , b , c {\displaystyle a,b,c} respectively, and medians m

1386-400: The median on the hypotenuse—that is, it goes through both the right-angled vertex and the midpoint of the side opposite that vertex. This is because the right triangle's orthocenter, the intersection of its altitudes, falls on the right-angled vertex while its circumcenter, the intersection of its perpendicular bisectors of sides , falls on the midpoint of the hypotenuse. In any right triangle

1428-409: The pre-pressurized hydrogen gas that fills the other end of the pump tube. As the piston accelerates toward the junction point, it rapidly compresses the hydrogen gas in the pump tube to a pressure of 60,000 psi (4,100 atm). The small projectile, meanwhile, rests in the adjacent depressurized launch tube. As the hydrogen gas reaches maximum pressure, a coupling holding the projectile in place

1470-434: The project continued, there were plans to elevate the tube and begin space launch trials potentially reaching speeds of up to 7 km/s (16,000 mph), or about Mach 21. The tests were designed as a precursor to the " Jules Verne Launcher ," an even larger light-gas gun with a 3,500 m (11,500 ft) barrel length designed in the early 1990s for first-stage satellite launch. This was to cost $ 1 billion, but funding

1512-490: The right angle basic to trigonometry. The meaning of right in right angle possibly refers to the Latin adjective rectus 'erect, straight, upright, perpendicular'. A Greek equivalent is orthos 'straight; perpendicular' (see orthogonality ). A rectangle is a quadrilateral with four right angles. A square has four right angles, in addition to equal-length sides. The Pythagorean theorem states how to determine when

1554-409: The right-angled vertex. The radius of the incircle of a right triangle with legs a {\displaystyle a} and b {\displaystyle b} and hypotenuse c {\displaystyle c} is The radius of the circumcircle is half the length of the hypotenuse, Thus the sum of the circumradius and the inradius is half the sum of the legs: One of

1596-403: The side adjacent to angle B {\displaystyle B} and opposite (or opposed to ) angle A , {\displaystyle A,} while side b {\displaystyle b} is the side adjacent to angle A {\displaystyle A} and opposite angle B . {\displaystyle B.} Every right triangle is half of

1638-546: The sides labeled opposite, adjacent and hypotenuse with reference to this angle according to the definitions above. These ratios of the sides do not depend on the particular right triangle chosen, but only on the given angle, since all triangles constructed this way are similar . If, for a given angle α, the opposite side, adjacent side and hypotenuse are labeled O , {\displaystyle O,} A , {\displaystyle A,} and H , {\displaystyle H,} respectively, then

1680-526: The trigonometric functions are For the expression of hyperbolic functions as ratio of the sides of a right triangle, see the hyperbolic triangle of a hyperbolic sector . The values of the trigonometric functions can be evaluated exactly for certain angles using right triangles with special angles. These include the 30-60-90 triangle which can be used to evaluate the trigonometric functions for any multiple of 1 6 π , {\displaystyle {\tfrac {1}{6}}\pi ,} and

1722-400: The vertical perpendicular to a horizontal base line. Closely related and important geometrical concepts are perpendicular lines, meaning lines that form right angles at their point of intersection, and orthogonality , which is the property of forming right angles, usually applied to vectors . The presence of a right angle in a triangle is the defining factor for right triangles , making

Super High Altitude Research Project - Misplaced Pages Continue

1764-584: Was not forthcoming and the project was eventually canceled in 1995. However, the SHARP gun continued to be used for high-speed tests in other areas of research, such as scramjet development. The concept of ballistic escape velocity is well proven. The largest challenge is maintaining such high velocities, because air resistance and aerothermal heating will significantly slow down any such object. Right angle Right Interior Exterior Adjacent Vertical Complementary Supplementary Dihedral In geometry and trigonometry ,

#82917