Circle Constructions - Student Guide - Part 2 PDF

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Summary

Notes for the year. Super helpful. fdjvkjfbvjkfbvkhfdb nmbckjbefkhvcfhkcvdhxdsvhcdvcjdvchjvfhcjvdfhhjfgfcvfgcvvcvcvcghdvchjdwsjkmbsmjcjbewlkdhuerfchvcjhbxkjevrfhregfiruhifrejdlxdjjkbcfhvfhjcejhdvejhdvxgdvcfhcvrhgvcrhvcrercrrNotes for the year. Super helpful. fdjvkjfbvjkfbvkhfdb nmbckjbefkhvcfhkcvdhx...

Description

Geometric Constructions Geometric constructions date back thousands of years to when Euclid, a Greek mathematician known as the “Father of Geometry,” wrote the book Elements. In Elements, Euclid formulated the five postulates that form the base for Euclidean geometry. To create all the figures and diagrams, Euclid used construction techniques extensively. A compass and straightedge are used to create constructions. A compass is used to draw circles or arcs and a straightedge is used to draw straight lines.

As you complete the task, keep these questions in mi How do you perform constructions related to circles used to justify these constructions?

eorems and explanations can be

In this task, you will apply what you have learned in this lesson to answer these questions.

Directions Complete each of the following tasks, reading the directions carefully as you go. Be sure to show all work where indicated, including inserting images of constructions created using the tool. If you are unable to take and insert screenshots of the construction tool, print this activity sheet and create the constructions by hand using a compass and straightedge. In addition to the answers you determine, you will be graded based on the work you show, or your solution process. So, be sure to show all your work and answer each question as you complete the task. Type all your work into this document so you can submit it to your teacher for a grade. You will be given partial credit based on the work you show and the completeness and accuracy of your explanations. Your teacher will give you further directions about how to submit your work. You may be asked to upload the document, e-mail it to your teacher, or print it and hand in a hard copy. Now, let’s get started!

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Student Guide (continued)

Step 1: Construct a circle through three points not on a line. a) Points D, E, and F are not in a line. To construct a circle through points D, E, and F, begin by drawing line segments𝐷𝐸 and 𝐸𝐹 . Then construct the perpendicular bisectors of 𝐷𝐸 and 𝐸𝐹, and name the point of intersection of the perpendicular bisectors O. How do you know that point O is the center of the circle that passes through the three points? (10 points) Please see that , in order to draw a circumcircle we have to first draw the perpendicular bisectors of the sides. The point at which these perpendicular bisectors meet is called circumcenter and the it is the center of circumcircle passes through three vertices D, E and F.

This is because, When we draw the circle from these vertices, we can see that the three sides forms that three chords of the circle. And circle has a property that , the perpendicular bisectors of chords always passes through the center. And here o is the point through which all the perpendicular bisectors of all three chords are passing through. Hence 0 is the center

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Student Guide (continued)

Step 2: Construct regular polygons inscribed in a circle. a) While constructing an equilateral triangle or a regular hexagon inscribed in a circle, you may have noticed that several smaller equilateral triangles are formed, like △PQR shown in the figure below. Explain why △PQR is an equilateral triangle. (5 points)

ΔPQR is an equilateral triangle because its sides are equal. In a regular hexagon, the r1 or radius of the incircle is equal to the side length of the hexagon.Segment RQ is the side length of a hexagon that will be inscribed in circle Q. Segment RP is the side length of a hexagon that will be inscribed in circle P. Both circles share segment PQ which is its radius. Since the radius of the incircle is equal to the side length of the hexagon, all sides of the triangle are equal, creating an equilateral triangle.

b) The completed construction of a regular hexagon is shown below. Explain why △ACF is a 30º-60º-90º triangle. (10 points)

You can prove that angle C is 30°. This is because we know that angle B is 120° also, and cointerior angles add to 180°, which makes angle BCF 60°. Then we can divide this by 2 as angle BCF is bisected by AC, so therefore angle ACF is 30° as 60 ÷ 2 = 30.You can then prove angle A is 90°. First we can find angle BAC because it is part of a triangle, and angles in a triangle add to 180°. This means we can do 30 + 120 = 150, and 180 150 = 30° which is angle BAC. Then we do 120 - 30 = 90, as the whole angle BAF is 120°. Copyright © Edgenuity Inc.

Student Guide (continued)

c) If you are given a circle with center C, how do you locate the vertices of a square inscribed in circle C? (5 points) The square is inscribed in a circle that means the square is drawn inside a circle such that it's vertices lie on the circle.You need to remember that the vertices of the square lie on the circumference of circle and the center of circle is also the midpoint of diagonals of the square i.e. Diagonal of the square is equal to the diameter of the circle.

Step 3: Construct tangent lines to a circle. a) 𝐽𝐿 is a diameter of circle K. If tangents to circle K are constructed through points L and J, what relationship would exist between the two tangents? Explain. (5 points) i. Let t be the line tangent at point J. We know that a tangent line at a point on a circle, is perpendicular to the diameter comprising that certain point.So t is perpendicular to JL Copyright © Edgenuity Inc.

Student Guide (continued)

let l be the tangent line through L. Then l is perpendicular to JL ii. So t and l are 2 different lines, both perpendicular to line JL.2 lines perpendicular to a third line, are parallel to each other, so the tangents t and l are parallel to each other.

b) The construction of a tangent to a circle given a point outside the circle can be justified using the second corollary to the inscribed angle theorem. An alternative proof of this construction is shown below. Complete the proof. (5 points) Given: Circle C is constructed so that CD = DE = AD; 𝐶𝐴 is a radius of circle C. Prove: 𝐴𝐸 is tangent to circle C.

Reasons

Statements 1.

Circle C is constructed so that CD = DE = AD;

1.

Given

𝐶𝐴 is a radius of circle C. 2.

𝐶𝐷 ≅ 𝐷𝐸 ≅ 𝐴𝐷

2.

Definition of congruence

3.

△ACD is an isosceles triangle;

3.

Definition of isosceles triangle

4.

Sum of interior angles of a triangle is 180 degrees

△ADE is an isosceles triangle. 4.

𝑚∠𝐶𝐴𝐷 + 𝑚∠𝐷𝐶𝐴 + 𝑚∠𝐴𝐷𝐶 = 180°; 𝑚∠𝐷𝐴𝐸 + 𝑚∠𝐴𝐸𝐷 + 𝑚∠𝐸𝐷𝐴 = 180°

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Student Guide (continued)

5.

Angle CAD is congruent to angle DCA;

5.

Isosceles triangle theorem

Angle DAE is congruent to angle AED 6.

𝑚∠𝐶𝐴𝐷 = 𝑚∠𝐷𝐶𝐴; 𝑚∠𝐷𝐴𝐸 = 𝑚∠𝐴𝐸𝐷

6.

Definition of congruence

7.

𝑚∠𝐶𝐴𝐷 + 𝑚∠𝐶𝐴𝐷 + 𝑚∠𝐴𝐷𝐶 = 180°; 𝑚∠𝐷𝐴𝐸 + 𝑚∠𝐷𝐴𝐸 + 𝑚∠𝐸𝐷𝐴 = 180°

7.

Substitution property

8.

2(𝑚∠𝐶𝐴𝐷) + 𝑚∠𝐴𝐷𝐶 = 180°; 2(𝑚∠𝐷𝐴𝐸) + 𝑚∠𝐸𝐷𝐴 = 180°

8.

Addition

9.

𝑚∠𝐴𝐷𝐶 = 180° – 2(𝑚∠𝐶𝐴𝐷);

9.

subtraction

𝑚∠𝐸𝐷𝐴 = 180° – 2(𝑚∠𝐷𝐴𝐸)

10.

∠𝐴𝐷𝐶 and ∠𝐸𝐷𝐴 are a linear pair.

11.

Angle ADC and Angle EDA are supplementary

11.

Linear pair postulate

12.

𝑚∠𝐴𝐷𝐶 + 𝑚∠𝐸𝐷𝐴 = 180°

12.

Definition of supplementary angles

13.

180° – 2(𝑚∠𝐶𝐴𝐷) + 180° – 2(𝑚∠𝐷𝐴𝐸) = 180°

13.

Substitution property

14.

360° – 2(𝑚∠𝐶𝐴𝐷) – 2(𝑚∠𝐷𝐴𝐸) = 180°

14.

Addition

15.

– 2(𝑚∠𝐶𝐴𝐷) – 2(𝑚∠𝐷𝐴𝐸) =− 180°

15.

Subtraction property

16.

𝑚∠𝐶𝐴𝐷 + 𝑚∠𝐷𝐴𝐸 = 90°

16.

Definition of complementary angles

17.

𝑚∠𝐶𝐴𝐷 + 𝑚∠𝐷𝐴𝐸 = 𝑚∠𝐶𝐴𝐸

17.

Angle addition postulate

18.

Angle CAE is 90 degrees

18.

Substitution property

19.

∠𝐶𝐴𝐸 is a right angle.

19.

Definition of right angle

20.

AE is perpendicular to AC

20.

Definition of perpendicular

21.

𝐴𝐸 is tangent to circle C.

21.

Converse of Radius-Tangent Theorem

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