Question

A regular polygon is shown, with one of its angle measures labeled a. 9 sided regular polygon with one angle labeled a If m∠a = (8z + 52)°, find the value of z. z = 128 z = 48 z = 16 z = 11 Question 8(Multiple Choice Worth 5 points) (Polygon Angle Sums LC) What is the sum of all interior angles of a 35-sided regular polygon? 6,120° 5,940° 180° 170° Question 9(Multiple Choice Worth 5 points) (Interior and Exterior Angles LC) One interior angle of a triangle is 45°, and the other two angles are congruent. Choose the equation that could be used to determine the degree measure of one of the congruent angles. x + 45 = 180 x − 45 = 90 2x + 45 = 180 2x − 45 = 90 Question 10(Multiple Choice Worth 5 points) (Interior and Exterior Angles MC) In triangle XYZ, m∠Y = 43.85° and m∠Z = 38.6°. Determine the measure of the exterior angle to ∠X. 25.65° 41.25° 82.45° 97.55° Question 11(Multiple Choice Worth 5 points) (Angle Relationships MC) Angles X and Y are supplementary. Angle X measures 132.54°, and angle Y measures (m − 15)°. Find m∠Y. 47.46° 62.46° 85.08° 117.54° Question 12(Multiple Choice Worth 5 points) (Pythagorean Theorem MC) Determine which set of side measurements could be used to form a right triangle. 14, 5, 15 3, 4, 5 9, 14, 16 5, 2, 7 Question 13 (Essay Worth 12 points) (Pythagorean Theorem and the Coordinate Plane HC) A map of three public schools was created using a coordinate plane where the origin represents the center of the town. Euclid Elementary School is graphed at (−4, 1), Math Middle School is graphed at (1, 1), and Hypotenuse High School is graphed at (−4, −3). Each unit on the graph represents 1 mile. Part A: Find the shortest distance, in miles, from Euclid Elementary School to Math Middle School. Show every step of your work. (2 points) Part B: Find the shortest distance, in miles, from Euclid Elementary School to Hypotenuse High School. Show every step of your work. (2 points) Part C: Find the shortest distance, in miles, from Math Middle School to Hypotenus

Answer 1

The correct responses are as follows;

First part; a = 11

Question 8; 5,940°

Question 9; 2·x + 45 = 180°

Question 10; 82.45°

Question 11; 47.46°

Question 12; 3, 4, and 5

Question 13;

Part A; 5 miles

Part B; 4 miles

Part C; miles

The details of the method used to arrive at the above responses are presented as follows;

The measure of an interior angle of a regular 9 sided polygon is 140°, therefore;

m∠a = (8·z + 52)°

m∠a = 140°

(8·z + 52)° = 140°

z = (140 - 52)/8

(140 - 52)/8 = 11

a = 11

Question 8: The sum of all interior angles of a 35-sided regular polygon is 5,940°. This is because the formula for the sum of interior angles of a polygon with n sides is (n - 2) × 180°.

(35 - 2) × 180 = 5940

Question 9: The equation that could be used to determine the degree measure of one of the congruent angles is 2·x + 45 = 180. This is because the sum of the angles of a triangle is 180°, and since one angle is 45° and the other two are congruent, we can write 45 + x + x = 180 and simplify it to 2·x + 45 = 180.

Question 10: The measure of the exterior angle to ∠X is 82.45°. This is because the exterior angle of a triangle is equal to the sum of the opposite interior angles, and since m∠Y = 43.85° and m∠Z = 38.6°, we can write 43.85 + 38.6 = 82.45.

Question 11: The value of m∠Y is 47.46°. This is because angles X and Y are supplementary, which means they add up to 180°, and since m∠X = 132.54° and m∠Y = (m - 15)°, we can write 132.54 + (m - 15) = 180 and solve for m to get m = 62.46. Then, we can substitute m into m∠Y to get (62.46 - 15)° = 47.46°.

Question 12: The set of side measurements that could be used to form a right triangle is 3, 4, 5. This is because they satisfy the Pythagorean theorem, which states that the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides, and since 3² + 4² = 5², they form a right triangle.

Question 13: Part A: The shortest distance, in miles, from Euclid Elementary School to Math Middle School is 5. This is because we can use the distance formula, which is derived from the Pythagorean theorem, to find the length of the line segment that connects the two schools on the coordinate plane.

The distance formula is d = √((x₂ - x₁)² + (y₂ - y₁)²), where (x₁, y₁) and (x₂, y₂) are the coordinates of the two points. Since Euclid Elementary School is at (−4, 1) and Math Middle School is at (1, 1), we can plug in the values and get d = √((1 - (-4))² + (1 - 1)²) = √(25 + 0) = √(25) = 5.

Part B: The shortest distance, in miles, from Euclid Elementary School to Hypotenuse High School is 4. This is because we can use the same distance formula as in Part A, but with different coordinates. Since Euclid Elementary School is at (−4, 1) and Hypotenuse High School is at (−4, −3), we can plug in the values and get d = √((-4 - (-4))² + (-3 - 1)²) = √(0 + 16) = √(16) = 4.

Part C: The shortest distance, in miles, from Math Middle School to Hypotenuse High School is about 6.4. This is because we can use the same distance formula as in Part A, but with different coordinates. Since Math Middle School is at (1, 1) and Hypotenuse High School is at (−4, −3), we can plug in the values and get d = √((-4 - 1)² + (-3 - 1)²) = √(25 + 16) = √(41) ≈ 6.4