Геометрія 8 клас
Table of Contents
- Чотирикутники
- Isosceles Trapezoid Construction Template
- Rhombus Template with Investigation Questions
- Подібність трикутників
- Similar Triangles Investigations
- Розв'язування прямокутних трикутників
- Applying Pythagoras Theorem
- Многокутники та їх площі
- Triangle Area Action!!! (V2)
Isosceles Trapezoid Construction Template
The quadrilateral below is said to be an ISOSCELES TRAPEZOID. Move points A & B, the two smaller points on the bases of this trapezoid, and the white points outside the trapezoid all around. Then answer the questions that follow.
Without Googling, how would you define an [b]isosceles trapezoid[/b]? Complete the following sentence definition: [br][br]An [b]isosceles trapezoid[/b] is ...
Use the tools below to construct a quadrilateral that always remains an ISOSCELES TRAPEZOID (no matter where you drag its vertices)!
Rename two of the vertices you have here so that the isosceles trapezoid you created would be named [i]ABCD[/i].
At this point, right click on the dashed line and hide it. Use the tools of GeoGebra to construct and measure sides, angles, and diagonals. [br][br]After doing so, be sure to plot the point at which the diagonals intersect. Then hide the diagonals and construct the four smaller non-overlapping triangles that would be formed and measure their side lengths and interior angle measures. [br][br]Which of the following statements are true for an isosceles trapezoid?
Similar Triangles Investigations
Similar Triangles Investigation 1
Similar Triangle Investigation 2
Applying Pythagoras Theorem
1. Introduction
[justify]The Pythagoras theorem is one of the most known results in mathematics and also one of the oldest known. There are hundreds of demonstrations of this theorem.[br][br]The understanding of the theorem is easy and has numerous applications in real life, as we will see in exercises in this section. It also has uses in advanced mathematics as well (vectorial analysis, functional analysis...)[/justify]
2. The Theorem
Given a right triangle with sides [i]a[/i] and [i]b[/i] and a hypotenuse [i]h[/i] (the side opposite the right angle). Then,[br][right][/right][img]https://www.matesfacil.com/pitagoras/teorema-Pitagoras.png[/img][br][br]Remember that...[br][list][br][*] triangle is a [b]right-angled triangle[/b] because it has a right angle, an angle of 90º or π / 2 radians[br][br][*] the [b]hypotenuse[/b] is the opposite side as the right angle.[br][/list][br][br][b]Note:[/b] [i]h[/i] is always bigger than the other sides, as shows [i]h > a[/i] and [i]h > b[/i].
3. Applying the Pythagoras Theorem
[b]Problem 1: [/b]Calculate the hypotenuse of the triangle with sides of 3cm and 4 cm.[br][br]Solution: [br][br]The sides are [math]a=3[/math] and [math]b=4[/math].[br][br]Applying the Pythagoras Theorem,[br][br][img]https://www.matesfacil.com/pitagoras/pitagoras1-1.png[/img][br][br]Therefor, the hypotenuse measures 5cm.[br][br][b]Problem 2: [/b]If the hypotenuse of a triangle measures 2 cm and one of it's sides measures 1cm, What does the other side measure?[br][br]Solution: [br][br]We call the sides [i]a[/i] and [i]b[/i], and the hypotenuse [i]h[/i].[br][br]We know that [math]h=2[/math] and [math]a=1[/math].[br][br]Because of Pythagoras we know that[br][br][math]h^2=a^2+b^2[/math][br][br]Substituting the values we know we have the following equation[br][br][img]https://www.matesfacil.com/pitagoras/pitagoras2-1.png[/img][br][br]Now we isolate [i]b[/i] from the equation:[br][br][img]https://www.matesfacil.com/pitagoras/pitagoras2-2.png[/img][br][br]We have written the positive and negative sign because in theory it's what we must do. But because [i]b[/i] represents a measurement, it can't be a negative number.[br][br]Therefor, the side measures[br][br][img]https://www.matesfacil.com/pitagoras/pitagoras2-3.png[/img][br][br]We can leave the square root or reduce it.
4. More Resolved Problems
Link: [url=https://www.matesfacil.com/english/secondary/Pythagoras-Theorem-Resolved-Problems.html][b]Resolved Problems:[/b] applying the Pythagoras Theorem[/url]
Triangle Area Action!!! (V2)
In the applet below, [color=#38761d][b]a triangle is shown[/b][/color]. [br][br]You can change the sizes of the [b][color=#1e84cc]blue angle[/color][/b] and [color=#ff00ff][b]pink angle[/b][/color] by using the sliders in the lower-left corner. You can also move the white points anywhere you'd like at any time. [br][br]Interact with the applet below for a few minutes.[br]Then answer the questions that follow.
1.
In this applet's dynamics, the triangle was transformed into another figure. What kind of figure is it? How does the applet imply your classification is correct?
2.
How does the [b]area of the newer figure[/b] compare with the [b]area of the original triangle[/b]?
3.
How does the [b]BASE [/b]of the newer figure compare with the [b]BASE[/b] of the original triangle?
4.
How does the [color=#9900ff][b]HEIGHT[/b][/color] of the newer figure compare with the [color=#444444][b]HEIGHT[/b][/color] of the original triangle?
5.
Given your responses to (1) - (4) above, describe how you can find the area of [b][color=#38761d]ANY TRIANGLE. [/color][/b]