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Common Core: High School - Geometry : High School: Geometry

Study concepts, example questions & explanations for Common Core: High School - Geometry

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All Common Core: High School - Geometry Resources

6 Diagnostic Tests 114 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #4 : Derive The Formula A = 1/2 Ab Sin(C) For The Area Of A Triangle By Drawing An Auxiliary Line From A Vertex Perpendicular To The Opposite Side

Stephanie is building a triangular garden in her backyard. She has cut wood for the sides with lengths of 5, 4, and 3 feet. Knowing that the height from the vertex $A$ to the base formed by the line $CB$ is 1.778 feet, first find the area that will be in the garden and then calculate each angle that Stephanie will have to put the boards together in order for the lengths to fit just right. Round your answer to the third decimal place if needed.

Screen shot 2020 08 12 at 9.51.18 am

Possible Answers:

$A = 5.334, \angle A = 117.263, \angle B = 36.346, \angle C = 26.391$

$A = 5.334, \angle A = 110, \angle B = 45, \angle C = 25$

$A = 6.12, \angle A = 110, \angle B = 45, \angle C = 25$

$A = 6.12, \angle A = 117.263, \angle B = 36.346, \angle C = 26.391$

Correct answer:

$A = 5.334, \angle A = 117.263, \angle B = 36.346, \angle C = 26.391$

Explanation:

We must begin by finding the area first. To find the area we begin by using our equation $A = \frac{1}{2}ab(sin(C))$ and recall that $sin(C) = \frac{h}{b}$ . We will let:

a = 6

b = 4

$A = \frac{1}{2}ab(sin(C))$

$A = \frac{1}{2}ab\frac{h}{b} \text{because $sin(C) = \frac{h}{b}$}$

$A = \frac{1}{2}ah$

$A = \frac{1}{2}(6)(1.778)$

A = 5.334

So our area is 5.34ft^2

How to find $\angle C$ : a=6 , b=4 , A=5.334

$A = \frac{1}{2}ab(sin(C))$

2A = ab(sin(C))

$\frac{2A}{ab} = sin(C)$

$sin^{-1}(\frac{2A}{ab}) = C$

$sin^{-1}(\frac{2(5.334)}{(6)(4)} = C$

$sin^{-1}(\frac{10.668}{24} = C$

26.391 = C

How to find $\angle B$ : a=6 , b=3 , A=5.334

$A = \frac{1}{2}ab(sin(C))$

2A = ab(sin(C))

$\frac{2A}{ab} = sin(C)$

$sin^{-1}(\frac{2A}{ab}) = C$

$sin^{-1}(\frac{2(5.334)}{(6)(3)}) = C$

36.346 = C

To find $\angle A$ , we know the angles of a triangle should add up to 180 so:

$180 = \angle A + \angle B + \angle C$

$180 = 26.391 + 36.346 + \angle C$

$117.263 = \angle C$

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Example Question #5 : Derive The Formula A = 1/2 Ab Sin(C) For The Area Of A Triangle By Drawing An Auxiliary Line From A Vertex Perpendicular To The Opposite Side

Using the triangle below, find the area then solve for .

Screen shot 2020 08 13 at 8.23.50 am

Possible Answers:

A = 22, h = 6

A = 22, h = 4

A = 24, h = 6

A = 24, h = 4

Correct answer:

A = 24, h = 4

Explanation:

To solve for , we first need to find the area using our formula $A = \frac{1}{2}ab(sin(C))$ , considering the triangle ACD . We will begin by labeling our variables:

a = 12

b = 8

C = 30

To find the area, we plug the values for the variables into the formula.

$A = \frac{1}{2}(12)(8)sin(30)$

A = 24

Knowing that our area is 24, we can now solve for . Recall that:

$sin(C) = \frac{h}{b}$

$\Rightarrow bsin(c) = h$

It follows that:

$A = \frac{1}{2}ab(sin(C))$

$A = \frac{1}{2}ah$ (because bsin(C) = h )

$24 = \frac{1}{2}(12)h$ (plugging in our values for area and )

24 = (6)h

4 = h

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Example Question #6 : Derive The Formula A = 1/2 Ab Sin(C) For The Area Of A Triangle By Drawing An Auxiliary Line From A Vertex Perpendicular To The Opposite Side

Complete the statement by choosing from one of the following

Regarding the triangle below, to find the area using the formula $A = \frac{1}{2}ab(sin(C))$ , I assume that sin(C) = ______

Screen shot 2020 08 19 at 4.39.16 pm

Possible Answers:

$\frac{h}{B}$

$\frac{b}{h}$

$\frac{h}{c}$

$\frac{h}{b}$

Correct answer:

$\frac{h}{b}$

Explanation:

Recall that $sin(\theta) = \frac{opposite}{hypotenuse}$ for right triangles. When we draw a line down from the vertex to the opposite side this creates two right triangles within the original obtuse triangle. If we are solving for sin(C) , then becomes our hypotenuse and becomes the opposite side we are working with. Therefore $sin(C) = \frac{h}{b}$ .

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Example Question #1 : Derive The Formula A = 1/2 Ab Sin(C) For The Area Of A Triangle By Drawing An Auxiliary Line From A Vertex Perpendicular To The Opposite Side

Using the information from the following triangle, find the area. Round to the second decimal place.

Screen shot 2020 08 19 at 4.48.00 pm

Possible Answers:

A = 31. 64

A = 29.49

A = 30.64

A = 25.67

Correct answer:

A = 30.64

Explanation:

Using the formula $A = \frac{1}{2}ab(sin(C))$ , we will let a = 8, b = 10 and C = 130

$A = \frac{1}{2}ab(sin(C))$

$A = \frac{1}{2}(8)(10)sin(130)$

A = 30.64

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Example Question #2 : Derive The Formula A = 1/2 Ab Sin(C) For The Area Of A Triangle By Drawing An Auxiliary Line From A Vertex Perpendicular To The Opposite Side

Given the triangle below, find the area and round to the second decimal place (in degrees).

Screen shot 2020 08 20 at 8.46.15 am

Possible Answers:

A = 12.49

A = 20.19

A = 15.32

A = 18.39

Correct answer:

A = 18.39

Explanation:

Screen shot 2020 08 20 at 8.46.52 am

Following from the figure above, we draw an auxiliary line from the vertex perpendicular to the opposite side. We will call this line for height and will label the new vertex . The formula we will use is $A = \frac{1}{2}ab(sin(C))$ . First we begin by labeling our variables. We will let be the base of our triangle and be the hypotenuse of the triangle formed by BCD . The values for the variables are as follows:

a=8

b=6

C=50

Plugging these into our formula, it follows that:

$A = \frac{1}{2}(8)(6)sin(50)$

A = 18.39

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Example Question #1 : Derive The Formula A = 1/2 Ab Sin(C) For The Area Of A Triangle By Drawing An Auxiliary Line From A Vertex Perpendicular To The Opposite Side

You are making your friend a triangular fabric decoration for their home. It has ripped so you need to go buy new fabric. The fabric you are buying is $3 per square foot. Using the dimensions below, how much money do you need to buy enough fabric?

Screen shot 2020 08 20 at 11.11.05 am

Possible Answers:

$A = \$156$

$A = \$540$

$A = \$162$

$A = \$324$

Correct answer:

$A = \$162$

Explanation:

Using the formula $A = \frac{1}{2}ab(sin(C))$ , we will let a=18 and b=10 .

$A = \frac{1}{2}ab(sin(C))$

$A = \frac{1}{2}ab\frac{h}{b}$ (because $sin(C) = \frac{h}{b}$ )

$A = \frac{1}{2}(18)(10)\frac{6}{10}$

A = 54

Then we must account for each square foot costing $3.

54*3 = 162

The answer is $162.

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Example Question #1 : Drawing Transformed Figures: Ccss.Math.Content.Hsg Co.A.5

If a triangle is in quadrant three and undergoes a transformation that moves each of its coordinate points to the left three units and down one unit, what transformation has occurred?

Possible Answers:

Translation

Extension

None of the other answers.

Rotation

Reflection

Correct answer:

Translation

Explanation:

To determine the type of transformation that is occurring in this particular situation, first recall the different types of transformations.

Translation: To move an object from its original position a certain distance without changing the object in any other way.

Transformation: Refers to any of the four changes that can be done to an object geographically. Transformations include, reflections, translations, rotations, and resizing the object.

Reflection: To flip the orientation of an object over a specific line or function.

Rotation: To rotate an object either clockwise or counter clockwise around a center point.

Since each of the triangle's coordinates is moved to the left and down, it is seen that the size and shape of the triangle remains the same but its location is different. Therefore, the transformation the triangle has undergone is a translation.

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Example Question #191 : Congruence

If a rectangle has the coordinate values, (a,b) , (c,b) , (a,d) , and (c,d) and after a transformation results in the coordinates (-a,b) , (-c, b) , (-a,d) , and (-c,d) identify the transformation.

Possible Answers:

Dilation

None of the other answers.

Extension

Rotation

Reflection

Correct answer:

Reflection

Explanation:

"If a rectangle has the coordinate values, (a,b) , (c,b) , (a,d) , and (c,d) and after a transformation results in the coordinates (-a,b) , (-c, b) , (-a,d) , and (-c,d) identify the transformation."

A transformation that changes the values by multiplying them by negative one is known as a reflection across the -axis or the line x=0

Therefore, this particular rectangle is being reflected across the -axis because the opposite of all the values have been taken.

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Example Question #2 : Draw Generalized Transformed Figures

If a rectangle has the coordinate values, (a,b) , (c,b) , (a,d) , and (c,d) and after a transformation results in the coordinates (a,b+2) , (c, b+2) , (a,d+2) , and (c,d+2) identify the transformation.

Possible Answers:

None of the other answers.

Dilation

Reflection

Translation

Rotation

Correct answer:

Translation

Explanation:

To identify the transformation that is occurring in this particular problem, recall the different transformations.

Translation: To move an object from its original position a certain distance without changing the object in any other way.

Transformation: Refers to any of the four changes that can be done to an object geographically. Transformations include, reflections, translations, rotations, and resizing the object.

Reflection: To flip the orientation of an object over a specific line or function.

Rotation: To rotate an object either clockwise or counter clockwise around a center point.

Looking at the starting and ending coordinates of the rectangle,

(a,b) , (c,b) , (a,d) , and (c,d) to (a,b+2) , (c, b+2) , (a,d+2) , and (c,d+2)

Since all the coordinates are increasing by two this is known as a translation.

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Example Question #3 : Draw Generalized Transformed Figures

If a rectangle has the coordinate values, (a,b) , (c,b) , (a,d) , and (c,d) and after a transformation results in the coordinates (a,-b) , (c, -b) , (a,-d) , and (c,-d) identify the transformation.

Possible Answers:

Extension

Reflection

Dilation

None of the others

Rotations

Correct answer:

Reflection

Explanation:

"If a rectangle has the coordinate values, (a,b) , (c,b) , (a,d) , and (c,d) and after a transformation results in the coordinates (a,-b) , (c, -b) , (a,-d) , and (c,-d) identify the transformation."

A transformation that changes the values by multiplying them by negative one is known as a reflection across the -axis or the line y=0

Therefore, this particular rectangle is being reflected across the -axis because the opposite of all the values have been taken.

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Common Core: High School - Geometry : High School: Geometry

Study concepts, example questions & explanations for Common Core: High School - Geometry

All Common Core: High School - Geometry Resources

Example Questions

Example Question #4 : Derive The Formula A = 1/2 Ab Sin(C) For The Area Of A Triangle By Drawing An Auxiliary Line From A Vertex Perpendicular To The Opposite Side

Example Question #5 : Derive The Formula A = 1/2 Ab Sin(C) For The Area Of A Triangle By Drawing An Auxiliary Line From A Vertex Perpendicular To The Opposite Side

Example Question #6 : Derive The Formula A = 1/2 Ab Sin(C) For The Area Of A Triangle By Drawing An Auxiliary Line From A Vertex Perpendicular To The Opposite Side

Example Question #1 : Derive The Formula A = 1/2 Ab Sin(C) For The Area Of A Triangle By Drawing An Auxiliary Line From A Vertex Perpendicular To The Opposite Side

Example Question #2 : Derive The Formula A = 1/2 Ab Sin(C) For The Area Of A Triangle By Drawing An Auxiliary Line From A Vertex Perpendicular To The Opposite Side

Example Question #1 : Derive The Formula A = 1/2 Ab Sin(C) For The Area Of A Triangle By Drawing An Auxiliary Line From A Vertex Perpendicular To The Opposite Side

Example Question #1 : Drawing Transformed Figures: Ccss.Math.Content.Hsg Co.A.5

Example Question #191 : Congruence

Example Question #2 : Draw Generalized Transformed Figures

Example Question #3 : Draw Generalized Transformed Figures

All Common Core: High School - Geometry Resources

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