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# Triangles

A triangle is a basic polygon used in geometry which consists of three straight lines and three angles. A triangle also possesses three vertices (corners). Commonly, there are three terms used for classifying triangles and this terminology is based on the lengths of a triangle’s sides:

1. Equilateral Triangle: This type of triangle has 3 equal sides and three angles all of 60 degrees.
2. Isosceles Triangle: This triangle has 2 sides of equal length and these two sides of the same length also have angles of the same value. These angles are opposite to these two sides.
3. Scalene Triangle: A scalene triangle has no sides or angles of the same value.

In addition to describing triangles by their sides, they can also be classified by their internal angles:

1. Right Angled: This triangle has one angle equaling 90 degrees.
2. Acute Triangle: All angles are less than 90 degrees.
3. Oblique Triangle: Does not contain a 90 degree angle.
4. Obtuse Triangle: A triangle in which one of the angles is greater than 90 degrees.

Furthermore, there are two other concepts which are often used in the study of geometry when working with triangles. These terms are known as congruency and similarity. Triangles are described as being congruent when they have the exact same shape and size. These triangles can have all angles and sides being exactly equal, but only three of these six measurements need to be identical when claiming congruency.

Conversely, similar triangles typically have identical angle measurements and proportional sides. However, there are some theorems related to the concept of similarity which have slightly different conditions. ### Volume, Surface Area and Nets

Can someone complete these for my understanding. See attached for details Surface area is the measurement of then ______ of a 3-dimensional shape. The 2-dimensional image of a 3-dimensional shape is called a _______. How many rectangles are in the cereal box net?_______ Number each rectangle. Use the dimensions of th

### Triangle Coordinates After Reflection and Mapping

Triangle abc has coordinates a(-1,3), b(-6,5), and c(-4,7): 1- The coordinates of triangle a'b'c', the image of triangle abc after a reflection in the x-axis are a'------, ----- b'-----, ----- c'-----, -----? 2- The coordinates of triangle a''b''c'', the image of triangle abc after a reflection in the line y=x, are a''-----,

### Triangle of forces

Both ends of a rope of length 4.8m are attached to a horizontal beam, at points 4.2m apart. A chandelier hangs on the rope, 1.2m from one of its ends. See attached for diagram (a) Calculate the angles that the two sections of the rope make with the beam, in degrees correct to two decimal places. The chandelier has mass 1

### Geometry: Angles and Triangles

1. For the two acute angles, m1=6x -3° and m2 =x + 2°. Solve for x and the measure of each angle. State the ideas from the tool kit that justify the procedure you use. 2. Find the minimum and maximum limits for the length of the third side of a triangle if the other two sides and 83' and 117'.

### Constructing a Lattice from a Heron Triangle

The triangle (15,34, 35; 252) is the smallest acute Heron triangle indecomposable into two Pythagorean triangles. Realize it as a lattice triangle, with one vertex at the origin.

Find all Heron triangles with inradius 1, then all heron triangles with inradius 3/2.

### How to Find 5 Triangles with Integer Sides

Find five triangles with integers sides (without no commun divisors ) such that the median AD from angle A to side BC is m(a) = b + c - a where a, b , c are sides opposite to angles A, B, C respectively. The inner circle is tangent at point X to segment BD on side BC. So find the 5 triangles of integers sides (with no commun di

### Solving Triangles: How Far Apart Are the Girls?

A rescue Helicopter at an altitude of 200 m spots two lost girls at the same time. Xavia is at an angle of depression of 13 degree. Yelena is at an angle of depression of 13 degree. From a point on the ground directly below (B) the Helicopter, Yelena is at an angle bearing of 120 degree, Xavia is at a bearing of 240 degree. How

### Proof Involving Circumcenter of Inscribed Triangle

Prove the following theorem: If the circumcenter of a triangle lies on a median of that triangle, the triangle is isosceles. Write the given and prove parts based on the diagram (see the attached file).

### Proof involving congruent triangles

Write a two column proof for the following... Please see the attachment for the full problem description. Write a two column proof for the following: Given: M is the midpoint of ¯KP,(KP) ⃡∥(WT) ⃡, and ¯WT≅¯MP. Prove: R is the midpoint of ¯KT.

### Hyperbolic rectangle

Using the Universal Hyperbolic Theorem we can show that the sum of the angles of a quadrilateral will be less than 360 degrees in hyperbolic geometry. The formal definition of a rectangle is a quadrilateral with four right angles therefore this figure does not exist in hyperbolic geometry.

### Finding angle and length of a post out of alignment using Geometry.

A 5 foot post is supposed to be vertical, but it is 4 inches out of alignment at the top. A-What is the angle of lean, round to the nearest tenth of a degree? B-How high does it reach, round to the nearest hundredth of an inch?

### The law of sines

Two sides and an angle (SSA) of a triangle are given. Determine whether the given measurements produce one triangle, two triangles, or no triangle. Solve each triagle that results. a = 11, c = 7.8, A = 61degrees

### Can a right triangle be scalene, isosceles, or equilateral?

Please provide detailed answers. * Can a right triangle be scalene, isosceles, or equilateral? Why or why not? What are the advantages and disadvantages of classifying triangles by sides or angles? * Describe the relationship between a triangle's centroid and orthocenter. Do they need to be inside the triangle? * How wo

### Drawing a Koch Snowflake

Part I: Here we will look at the Koch snowflake which is constructed as follows: a. Start by drawing an equilateral triangle: we will call this T1 as it is the first step in the process. b. Next, on each of the three sides of the equilateral triangle, identify the middle third of that side, erase it, and then put an equi

### In a triangle the sum of the angles is 180 degrees. If Angle A is four times Angle B and Angle C is 27 degrees more than three times Angle A. Find the measures of the three angles to the nearest degree.

In a triangle the sum of the angles is 180 degrees. If Angle A is four times Angle B and Angle C is 27 degrees more than three times Angle A. Find the measures of the three angles to the nearest degree.

### Find the measures of the three angles.

In a triangle the sum of the angles is 180 degrees. If Angle A is four times Angle B and Angle C is 27 degrees more than three times Angle A. Find the measures of the three angles to the nearest degree.

### Euclidean three space

Suppose instead of working on the Euclidean plane we study geometry on a sphere in (Euclidean) three space. We interpret point to mean any point on the sphere and we interpret line to mean any great circle on the sphere (that is any circumference of the sphere). a. Is Euclidâ??s parallel postulate true in this setting? b. Ho

### Applying Geometric Theorem

Theorem: If a, b, and c are numbers for which the sum of any two of them is greater than the third, then there is a triangle whose sides are of length a, b, and c. Explain the following proof: Assume that a â?¥ b and a â?¥ c. Draw a line segment of length a. At one end draw a circle of radius b. At the other end dr

### Centroid of a Triangle

Using the fact that the centroid of a triangle lies at the intersection of the triangle's medians, which is the point that lies one-third of the way from each side toward the opposite vertex. Find the centroid of the triangle whose vertices are (0,0), (a,0) and (0,b). Assume a > 0 and b >0.

### Inequalities: triangle-ski ramp

Write an inequality in the variable x for the degree measure of the smallest angle of a triangle given that the degree measure of the smallest angle is at most 30 degrees diagram top angle x+8 bottom left x right angle has ?

### Geometry Supplementary Angles

1. Use the figure below to find the following: [Please refer to the attachment for the figure and problem details] 2. Angle PQR and Angle ABC are supplementary angles and Angle PQR is nine times as large as Angle ABC. Determine the measure of each angle. 3. What is the exact area of a circle with a diameter of

### Pascal's Triangle and the Binomial Theorem

1. Use Pascal's triangle to expand (x + y)6 2. For the expansion of (k + t)22 state: a) the number of terms b) the degree of each term c) the first four terms in the expansion, without coefficients d) the coeffcients of the first three terms 3. How is the sum of the entries in row 5 in Pascal's triangle re

### Pascal's Triangle: Five-Square by Five-Square Gameboard

If the numerical coefficients are disregarded, the following terms appear in the expansion of (x + y)11 State the value of k in each case: a) x2 yk b) xk y11. See the attached file.

### Similar Triangles and Finding Coordinates

Construct the quadrature of 3 squares. Let a = 3 be a line segment that extends along the y-axis from the origin O to the point B at (0,3). Let b = 4 be a line segment that extends along the x-axis from the origin O to the point A at (4,0). Let x be the hypotenuse of triangle ABO. Let c = 2 be a line segment extending above

### Geometry and Mensuration

Show all work step by step for the following problems. 1. Determine the volume and the surface area of a cylinder that has a height of 18 inches and a base of 6 inches. Use PI symbol whenever needed to round off your answer to the nearest hundredth. 2. Find the area of a triangle that has two sides with the measurements

### Combining Shapes

Need the following questions answered: 1. What two shapes can be combined to make a trapezoid? 2. List the shapes to use and the number of each to show 3 ways to make a hexagon. 3. What shapes can be use to make 2 triangles? 4. What shapes can be used to make 4 squares? 5. Draw 3 different shapes, then draw a shape yo

### Area of Shaded Region and Measurements

1. Find the area of the shaded region on the following geoboard if the unit of measurement is 1 cm2. 2. For each of the following, can the measures represent sides of a right triangle? Explain your answers.

### Area of a music theatre

Please show and explain your work as best you can with the keyboard even if it is a one-step process. The preferred seating area of a music theater has a base that measures 34 yd and a height of 39.6 yd. Find the area if the shape of the theater is A triangle A parallelogram A rectangle

### MTH 214

1. Draw at least 4 different nets that would form a cube. 2. Are angles of a triangle considered to be supplementary? Explain your answer. 3. Classify the following as true or false. Give Reasons in support of your answer. a. Two distinct lines that do not intersect are parallel b. No square is a