Problem 8
Answer: Sometimes true
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A rhombus is not always a square. Though it is possible for a rhombus to be a square. We can have the interior angles be non-right angles leading to a non-rectangular rhombus. If it said "a square is a rhombus" then the statement would always be true. Any square is a rhombus
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Problem 9
Answer: (3,-9)
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Translating "five units up and seven units to the left" means we have the rule (x,y) --> (x-7,y+5)
So we subtract 7 from the x coordinate: x = 10 becomes x-7 = 10-7 = 3
and we add 5 to the y coordinate: y = -14 becomes y+5 = -14+5 = -9
Paired up, the new point is (3,-9)
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Problem 10
Answer: 54
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This is a regular pentagon, so the five interior angles are congruent to each other. Similarly, the exterior angles are congruent to each other as well. Each exterior angle is E = 360/n = 360/5 = 72 degrees so each interior angle is I = 180-E = 180-72 = 108
Focus on the two triangles above the pentagon. They are congruent with the bottom acute angles x degrees. They each have a right angle and the missing angle is 90-x as it is complemtentary to angle x
Along the top we have the angles: 90-x, 108, and 90-x
See the attached figure for a visual
Add those three angles up and set the result equal to 180 degrees. Solve for x
(90-x)+(108)+(90-x) = 180
(-x-x)+(90+108+90) = 180
-2x+288 = 180
-2x+288-288 = 180-288
-2x = -108
-2x/(-2) = -108/(-2)
x = 54
Answer: Sometimes true
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A rhombus is not always a square. Though it is possible for a rhombus to be a square. We can have the interior angles be non-right angles leading to a non-rectangular rhombus. If it said "a square is a rhombus" then the statement would always be true. Any square is a rhombus
========================================================
Problem 9
Answer: (3,-9)
--------------------------------------
Translating "five units up and seven units to the left" means we have the rule (x,y) --> (x-7,y+5)
So we subtract 7 from the x coordinate: x = 10 becomes x-7 = 10-7 = 3
and we add 5 to the y coordinate: y = -14 becomes y+5 = -14+5 = -9
Paired up, the new point is (3,-9)
========================================================
Problem 10
Answer: 54
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This is a regular pentagon, so the five interior angles are congruent to each other. Similarly, the exterior angles are congruent to each other as well. Each exterior angle is E = 360/n = 360/5 = 72 degrees so each interior angle is I = 180-E = 180-72 = 108
Focus on the two triangles above the pentagon. They are congruent with the bottom acute angles x degrees. They each have a right angle and the missing angle is 90-x as it is complemtentary to angle x
Along the top we have the angles: 90-x, 108, and 90-x
See the attached figure for a visual
Add those three angles up and set the result equal to 180 degrees. Solve for x
(90-x)+(108)+(90-x) = 180
(-x-x)+(90+108+90) = 180
-2x+288 = 180
-2x+288-288 = 180-288
-2x = -108
-2x/(-2) = -108/(-2)
x = 54