Altitude
In mathematics, altitude refers to the perpendicular distance from a vertex of a triangle to the opposite side. It is used to calculate the area of a triangle and is a key concept in geometry. Altitude helps in understanding the relationship between the sides and angles of a triangle, and is essential for solving various geometric problems.
4 Key excerpts on "Altitude"
eBook - ePub- Mel Friedman(Author)
- 2012(Publication Date)
- Research & Education Association(Publisher)
...By the Pythagorean Theorem, the lengths of the three sides of a right triangle are related by the formula c 2 = a 2 + b 2 where c is the hypotenuse and a and b are the other two sides (the legs). The Pythagorean Theorem is discussed in more detail in the next section. An Altitude, or height, of a triangle is a line segment from a vertex of the triangle perpendicular to the opposite side. For an obtuse triangle, the Altitude sometimes is drawn as a perpendicular line to an extension of the opposite side. The area of a triangle is given by where h is the Altitude and b is the base to which the Altitude is drawn. A line segment connecting a vertex of a triangle and the midpoint of the opposite side is called a median of the triangle. A line that bisects and is perpendicular to a side of a triangle is called a perpendicular bisector of that side. An angle bisector of a triangle is a line that bisects an angle and extends to the opposite side of the triangle. The line segment that joins the midpoints of two sides of a triangle is called a midline of the triangle. An exterior angle of a triangle is an angle formed outside a triangle by one side of the triangle and the extension of an adjacent side. PROBLEM The measure of the vertex angle of an isosceles triangle exceeds the measure of each base angle by 30°. Find the value of each angle of the triangle. SOLUTION In an isosceles triangle, the angles opposite the congruent sides (the base angles) are, themselves, congruent and of equal value. Therefore, 1. Let x = the measure of each base angle 2. Then x + 30 = the measure of the vertex angle We can solve for x algebraically by keeping in mind that the sum of all the measures of the angles of a triangle is 180°. Therefore, the base angles each measure 50°, and the vertex angle measures 80°. THE PYTHAGOREAN THEOREM The Pythagorean Theorem pertains to a right triangle, which, as we saw, is a triangle that has one 90° angle...
eBook - ePub- Douglas K. Brumbaugh, Peggy L. Moch, MaryE Wilkinson(Authors)
- 2004(Publication Date)
- Routledge(Publisher)
...If you find it more comfortable to view an Altitude as a vertical segment, then you may want to rotate the page as you examine the four Altitudes of ABCD. Notice that a new point is created where each Altitude meets the opposite side (or extended side); for example, the height from Vertex B is perpendicular to at F. The sides of ABCD are congruent in pairs, thus you will see that the height of the polygon depends on your orientation as you view the sketch. Fig. 4.17. Your Turn 15. Write an informal definition for each term. a) Side of a polygon—How many sides does a 12-gon have? b) Vertices of a polygon—How many vertices does a 7-gon have? c) Diagonal of a polygon—How many diagonals does a 5-gon have? d) Altitude of a polygon—How many Altitudes does a 3-gon have? Circles Some people consider a circle to be a special polygon—with an infinite number of infinitely short sides. Others prefer to put the circle in a special category all its own. Because a circle is a simple closed curve, it divides the plane into three sets of points (inside the circle, outside the circle, and on the circle). When you used a piece of string to model the polygons in Table 4.1, did your model start to look a bit like a circle as you added more and more sides without changing the length of the string? It is easy to see the connection with polygons. However, as you look at circles, such as the one shown in Fig. 4.18, you do not see infinitely short sides, only smooth and perfectly rounded curves, making it easy to argue that circles should have their own category. Fig. 4.18. Sometimes people refer to circular pieces of material as circles—perhaps a coin, poker chip, or other disk. In fact, these items are right circular cylinders; they may be very short cylinders, but they are not figures in a plane and they have more dimensions than do circles. A circle is a point and every member of the set of points is the exact same distance from the center in the plane...
eBook - ePubGRE - Quantitative Reasoning
QuickStudy Laminated Reference Guide
- (Author)
- 2018(Publication Date)
- QuickStudy Reference Guides(Publisher)
...Geometry Topics include parallel and perpendicular lines, circles, triangles, quadrilaterals, other polygons, congruent and similar figures, three-dimensional figures, area, perimeter, volume, the Pythagorean theorem, and angle measurement in degrees. Angles Supplementary angles: The sum of the angles equals 180°. Complementary angles: The sum of the angles equals 90°. Points Point: An exact position or location on a plane surface. The location of a point is determined by its ordered pair, (x, y), where x determines the horizontal location and y determines the vertical location. EX: (2, 6), (-1, -9), and (0, 0) The distance between two points is found using this formula: d = √[(x 2 – x 1) 2 + (y 2 – y 1) 2 ] EX: Find the distance between the points (-1, -1) and (2, 3) Use the distance formula: d = √[(x 2 – x 1) 2 +. (y 2 – y 1) 2 ] Plug in the known values: d = √[(2 – -1) 2 + (3 – -1) 2 ] Subtract: d = √[(3) 2 + (4) 2 ] Apply the exponents: d = √[9 + 16] Add: d = √[25] Take the square root: d = 5 Lines A line has no width or curves and continues forever in two directions. A line is the shortest distance between two points; two points define a line. The equation of a line is y = mx + b, where m represents the slope of the line and b represents the y -intercept. The slope represents the direction of the line and can be found by using the change of y divided by the change of x, or rise over run. EX: What is the slope of the line that contains the points (1, -1) and (-2, 8)? (8 – -1) ÷ (-2 – 1) = 9 ÷ -3 = -3 Lines that rise as you go to the right have a positive...
- Rebecca Dayton(Author)
- 2013(Publication Date)
- Research & Education Association(Publisher)
...The two smaller triangles created are similar to each other and similar to the original triangle. Thus, Δ WZX ~ Δ YZW ~ Δ YWX Because the triangles are similar, the following theorems can be stated: 1) If an Altitude is drawn to the hypotenuse of a right triangle, then each leg is the geometric mean between the hypotenuse and its touching segment on the hypotenuse. 2) If an Altitude is drawn to the hypotenuse of a right triangle, then it is the geometric mean between the segments of the hypotenuse. Example: Find the value of x and y. To find the value of x, To find the value of y, Note: You will notice that y can be found using the Pythagorean Theorem: 6 2 + 9 2 = y 2. Special Right Triangles A 30°-60°-90° triangle is a type of special right triangle whose angles measure 30°, 60°, and 90°. The lengths of the sides of a 30°-60°-90° triangle are in a ratio of 1: : 2. Note: The shortest side is across from the smallest angle and the longest side is across from the largest angle. Note: The Altitude of an equilateral triangle forms two 30°-60°-90° triangles. A 45°-45°-90° triangle is a type of special right triangle whose angles measure 45°, 45°, and. 90°. The lengths of the sides of a 45°-45°-90° triangle are in a ratio of 1: 1:. Note: A 45°-45°-90° triangle has two sides of equal length; therefore, it is an isosceles right triangle. Example: Determine the value of x and y. Since the hypotenuse is 8, the side opposite the 30° angle must be. The length of the side opposite the 60° angle, ∠ A, is found by multiplying the length of the shortest side by...
