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Chapter 1: Problem 46

Show that the points form the vertices of the indicated polygon. Isosceles triangle: (2,3),(4,9),(-2,7)

Short Answer

Expert verified
Yes, the given points form the vertices of an isosceles triangle as two sides are of equal length.

Step by step solution

01

Calculation of distance between point (2,3) and (4,9)

Use the distance formula to find the length: \( \sqrt{(4-2)^2 + (9-3)^2} = \sqrt{2^2 + 6^2} = \sqrt{40} \)
02

Calculation of distance between point (2,3) and (-2,7)

Use the distance formula to find the length: \( \sqrt{(-2-2)^2 + (7-3)^2} = \sqrt{-4^2 + 4^2} = \sqrt{32} \)
03

Calculation of distance between point (4,9) and (-2,7)

Use the distance formula to find the length: \( \sqrt{(-2-4)^2 + (7-9)^2} = \sqrt{-6^2 + (-2)^2} = \sqrt{40} \)
04

Concluding the nature of the triangle

Compare the calculated lengths. If two of them are equal, the points form an isosceles triangle. In this case, the distances between the first and the third points are equal, hence the points form an isosceles triangle.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Isosceles Triangle
An isosceles triangle is a special kind of triangle where two of its sides are of equal length. This unique property makes it distinct from other types of triangles. The third side can be the longest or shortest, but two sides are always equally long.
When working with an isosceles triangle in geometry, one of the key insights is recognizing these equal sides. Not only does it help in classifying the triangle, but it also assists in solving geometric problems including finding missing angles or determining symmetry.
  • If you ever get a set of coordinates like (2,3), (4,9), and (-2,7) and need to figure out if they form an isosceles triangle, comparing the lengths between these points is crucial.
  • Always remember that in an isosceles triangle, because two sides are the same, the angles opposite those sides are also equal.
This property can be helpful in various geometry problems or proofs.
Exploring Coordinate Geometry
Coordinate geometry, also known as analytic geometry, merges algebra with geometry using a coordinate plane. It allows us to describe geometric figures numerically and solve geometric problems through algebraic processes.
In coordinate geometry, we use the coordinate system with the x-axis and y-axis to plot points and lines. This system helps us analyze shapes, sizes, and distances between points on the plane.
  • The points from our exercise are plotted as (2,3), (4,9), and (-2,7). To determine the type of triangle they form, we utilize the coordinate system effectively.
  • Using coordinates, you can calculate areas, perimeters, or angles of various shapes, but more importantly, you can visualize the problems at hand.
This tool is invaluable because it turns visual problems into numerical ones that can be easily tackled with algebraic methods.
Performing Distance Calculation
Distance calculation in coordinate geometry refers to finding the actual 'as-the-crow-flies' distance between two points. The distance formula is essentially a practical application of the Pythagorean theorem.
The formula is given by: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] where
  • \( (x_1, y_1) \) represent the coordinates of the first point, and
  • \( (x_2, y_2) \) represent the coordinates of the second point.
This formula calculates the linear distance between two coordinates on a plane by forming a right triangle and applying the theorem.
In the isosceles triangle exercise, distances calculated were:
  • Between (2,3) and (4,9) was \( \sqrt{40} \)
  • Between (2,3) and (-2,7) was \( \sqrt{32} \)
  • Between (4,9) and (-2,7) was \( \sqrt{40} \)
Recognizing these calculations as not just numbers, but segments of the triangle, allows us to conclude something about the shape they form. Two equal lengths indicate the isosceles nature.

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Most popular questions from this chapter

The lengths (in feet) of the winning men's discus throws in the Olympics from 1920 through 2008 are listed below. (Source: International Olympic Committee) $$\begin{array}{llllll} 1920 & 146.6 & 1956 & 184.9 & 1984 & 218.5 \\ 1924 & 151.3 & 1960 & 194.2 & 1988 & 225.8 \\ 1928 & 155.3 & 1964 & 200.1 & 1992 & 213.7 \\ 1932 & 162.3 & 1968 & 212.5 & 1996 & 227.7 \\ 1936 & 165.6 & 1972 & 211.3 & 2000 & 227.3 \\ 1948 & 173.2 & 1976 & 221.5 & 2004 & 229.3 \\ 1952 & 180.5 & 1980 & 218.7 & 2008 & 225.8 \end{array}$$ (a) Sketch a scatter plot of the data. Let \(y\) represent the length of the winning discus throw (in feet) and let \(t=20\) represent 1920 (b) Use a straightedge to sketch the best-fitting line through the points and find an equation of the line. (c) Use the regression feature of a graphing utility to find the least squares regression line that fits the data. (d) Compare the linear model you found in part (b) with the linear model given by the graphing utility in part (c). (e) Use the models from parts (b) and (c) to estimate the winning men's discus throw in the year 2012 .

Sketch the graph of the function. $$ g(x)=4 \llbracket x \rrbracket $$

Use the intercept form to find the equation of the line with the given intercepts. The intercept form of the equation of a line with intercepts \((a, 0)\) and \((0, b)\) is $$\frac{x}{a}+\frac{y}{b}=1, a \neq 0, b \neq 0.$$ \(x\) -intercept: (2,0) \(y\) -intercept: (0,3)

Identify any relationships that exist among the lines, and then use a graphing utility to graph the three equations in the same viewing window. Adjust the viewing window so that the slope appears visually correct- that is, so that parallel lines appear parallel and perpendicular lines appear to intersect at right angles. (a) \(y=x-8\) (b) \(y=x+1\) (c) \(y=-x+3\)

Determine whether the variation model is of the form \(y=k x\) or \(y=k / x,\) and find \(k .\) Then write \(a\) model that relates \(y\) and \(x\). $$ \begin{array}{|c|c|c|c|c|c|} \hline x & 5 & 10 & 15 & 20 & 25 \\ \hline y & 24 & 12 & 8 & 6 & \frac{24}{5} \\ \hline \end{array} $$
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