## 0024 Geometry

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## MTEL General Curriculum Mathematics Practice

 Question 1

#### What is the perimeter of a right triangle with legs of lengths x and 2x?

 A $$\large 6x$$Hint: Use the Pythagorean Theorem. B $$\large 3x+5{{x}^{2}}$$Hint: Don't forget to take square roots when you use the Pythagorean Theorem. C $$\large 3x+\sqrt{5}{{x}^{2}}$$Hint: $$\sqrt {5 x^2}$$ is not $$\sqrt {5}x^2$$. D $$\large 3x+\sqrt{5}{{x}^{{}}}$$Hint: To find the hypotenuse, h, use the Pythagorean Theorem: $$x^2+(2x)^2=h^2.$$ $$5x^2=h^2,h=\sqrt{5}x$$. The perimeter is this plus x plus 2x.
Question 1 Explanation:
Topic: Recognize and apply connections between algebra and geometry (e.g., the use of coordinate systems, the Pythagorean theorem) (Objective 0024).
 Question 2

#### The polygon depicted below is drawn on dot paper, with the dots spaced 1 unit apart.  What is the perimeter of the polygon?

 A $$\large 18+\sqrt{2} \text{ units}$$Hint: Be careful with the Pythagorean Theorem. B $$\large 18+2\sqrt{2}\text{ units}$$Hint: There are 13 horizontal or vertical 1 unit segments. The longer diagonal is the hypotenuse of a 3-4-5 right triangle, so its length is 5 units. The shorter diagonal is the hypotenuse of a 45-45-90 right triangle with side 2, so its hypotenuse has length $$2 \sqrt{2}$$. C $$\large 18 \text{ units}$$Hint: Use the Pythagorean Theorem to find the lengths of the diagonal segments. D $$\large 20 \text{ units}$$Hint: Use the Pythagorean Theorem to find the lengths of the diagonal segments.
Question 2 Explanation:
Topic: Recognize and apply connections between algebra and geometry (e.g., the use of coordinate systems, the Pythagorean theorem) (Objective 0024).
 Question 3

#### Tetrahedron

Hint:
All the faces of a tetrahedron are triangles.

#### Triangular Prism

Hint:
A prism has two congruent, parallel bases, connected by parallelograms (since this is a right prism, the parallelograms are rectangles).

#### Triangular Pyramid

Hint:
A pyramid has one base, not two.

#### Trigon

Hint:
A trigon is a triangle (this is not a common term).
Question 3 Explanation:
Topic: Classify and analyze three-dimensional figures using attributes of faces, edges, and vertices (Objective 0024).
 Question 4

#### An equiangular triangle that is not equilateral.

Hint:
The AAA property of triangles states that all triangles with corresponding angles congruent are similar. Thus all triangles with three equal angles are similar, and are equilateral.

#### An equiangular quadrilateral that is not equilateral.

Hint:
A rectangle is equiangular (all angles the same measure), but if it's not a square, it's not equilateral (all sides the same length).

#### An equilateral quadrilateral that is not equiangular.

Hint:
This rhombus has equal sides, but it doesn't have equal angles:

#### An equiangular hexagon that is not equilateral.

Hint:
This hexagon has equal angles, but it doesn't have equal sides:
Question 4 Explanation:
Topic: Classify and analyze polygons using attributes of sides and angles (Objective 0024).
 Question 5

#### 4 lines of reflective symmetry, 1 center of rotational symmetry.

Hint:
Try cutting out a shape like this one from paper, and fold where you think the lines of reflective symmetry are (or put a mirror there). Do things line up as you thought they would?

#### 2 lines of reflective symmetry, 1 center of rotational symmetry.

Hint:
Try cutting out a shape like this one from paper, and fold where you think the lines of reflective symmetry are (or put a mirror there). Do things line up as you thought they would?

#### 0 lines of reflective symmetry, 1 center of rotational symmetry.

Hint:
The intersection of the diagonals is a center of rotational symmetry. There are no lines of reflective symmetry, although many people get confused about this fact (best to play with hands on examples to get a feel). Just fyi, the letter S also has rotational, but not reflective symmetry, and it's one that kids often write backwards.

#### 2 lines of reflective symmetry, 0 centers of rotational symmetry.

Hint:
Try cutting out a shape like this one from paper. Trace onto another sheet of paper. See if there's a way to rotate the cut out shape (less than a complete turn) so that it fits within the outlines again.
Question 5 Explanation:
Topic: Analyze geometric transformations (e.g., translations, rotations, reflections, dilations); relate them to concepts of symmetry (Objective 0024).
 Question 6

#### Which of the following nets will not fold into a cube?

 A Hint: If you have trouble visualizing, cut them out and fold (during the test, you can tear paper to approximate). B C Hint: If you have trouble visualizing, cut them out and fold (during the test, you can tear paper to approximate). D Hint: If you have trouble visualizing, cut them out and fold (during the test, you can tear paper to approximate).
Question 6 Explanation:
Topic: Match three-dimensional figures and their two-dimensional representations (e.g., nets, projections, perspective drawings) (Objective 0024).
 Question 7

#### 2 pentagons and 5 rectangles.

Hint:
These can be assembled to form a pentagonal prism, not a pentagonal pyramid.

#### 1 square and 5 equilateral triangles.

Hint:
You need a pentagon for a pentagonal pyramid.

#### 1 pentagon and 10 isosceles triangles.

Question 7 Explanation:
Topic:Classify and analyze three-dimensional figures using attributes of faces, edges, and vertices (Objective 0024).
 Question 8

#### 4 congruent sides

Hint:
The most common definition of a rhombus is a quadrilateral with 4 congruent sides.

#### A center of rotational symmetry

Hint:
The diagonal of a rhombus separates it into two congruent isosceles triangles. The center of this line is a center of 180 degree rotational symmetry that switches the triangles.

#### 4 congruent angles

Hint:
Unless the rhombus is a square, it does not have 4 congruent angles.

#### 2 sets of parallel sides

Hint:
All rhombi are parallelograms.
Question 8 Explanation:
Topic: Classify and analyze polygons using attributes of sides and angles, and symmetry (Objective 0024).
 Question 9

#### A sphere

Hint:
All views would be circles.

#### A cone

Hint:
Two views would be triangles, not rectangles.

#### A pyramid

Hint:
How would one view be a circle?
Question 9 Explanation:
Topic: Match three-dimensional figures and their two-dimensional representations (e.g., nets, projections, perspective drawings) (Objective 0024).
 Question 10

#### In the triangle below, $$\overline{AC}\cong \overline{AD}\cong \overline{DE}$$ and $$m\angle CAD=100{}^\circ$$.  What is $$m\angle DAE$$?

 A $$\large 20{}^\circ$$Hint: Angles ACD and ADC are congruent since they are base angles of an isosceles triangle. Since the angles of a triangle sum to 180, they sum to 80, and they are 40 deg each. Thus angle ADE is 140 deg, since it makes a straight line with angle ADC. Angles DAE and DEA are base angles of an isosceles triangle and thus congruent-- they sum to 40 deg, so are 20 deg each. B $$\large 25{}^\circ$$Hint: If two sides of a triangle are congruent, then it's isosceles, and the base angles of an isosceles triangle are equal. C $$\large 30{}^\circ$$Hint: If two sides of a triangle are congruent, then it's isosceles, and the base angles of an isosceles triangle are equal. D $$\large 40{}^\circ$$Hint: Make sure you're calculating the correct angle.
Question 10 Explanation:
Topic: Classify and analyze polygons using attributes of sides and angles, including real-world applications. (Objective 0024).
 Question 11

#### If the polygon shown above is reflected about the y axis and then rotated 90 degrees clockwise about the origin, which of the following graphs is the result?

 A Hint: Try following the point (1,4) to see where it goes after each transformation. B C Hint: Make sure you're reflecting in the correct axis. D Hint: Make sure you're rotating the correct direction.
Question 11 Explanation:
Topic: Analyze and apply geometric transformations (e.g., translations, rotations, reflections, dilations); relate them to concepts of symmetry, similarity, and congruence; and use these concepts to solve problems (Objective 0024).
 Question 12

#### 7.5 meters

Hint:
Here is a picture, note that the large and small right triangles are similar:

One way to do the problem is to note that there is a dilation (scale) factor of 5 on the shadows, so there must be that factor on the heights too. Another way is to note that the shadows are twice as long as the heights.

Hint:
Draw a picture.

Hint:
Draw a picture.

#### 45 meters

Hint:
Draw a picture.
Question 12 Explanation:
Topic: Apply geometric transformations (e.g., translations, rotations, reflections, dilations); relate them to similarity, ; and use these concepts to solve problems (Objective 0024) . Fits in other places too.
 Question 13

#### A 90 degree clockwise rotation about (2,1) followed by a translation of two units to the right.

Hint:
Part of the figure would move below the x-axis with these transformations.

#### A translation 3 units up, followed by a reflection about the line y=x.

Hint:
See what happens to the point (5,1) under this set of transformations.

#### A 90 degree clockwise rotation about (2,1) followed by a translation of 2 units to the right.

Hint:
See what happens to the point (3,3) under this set of transformations.
Question 13 Explanation:
Topic:Analyze and apply geometric transformations (e.g., translations, rotations, reflections, dilations) (Objective 0024).
 Question 14

#### How many of the figures pictured above have at least one line of reflective symmetry?

 A $$\large 1$$ B $$\large 2$$Hint: The ellipse has 2 lines of reflective symmetry (horizontal and vertical, through the center) and the triangle has 3. The other two figures have rotational symmetry, but not reflective symmetry. C $$\large 3$$ D $$\large 4$$Hint: All four have rotational symmetry, but not reflective symmetry.
Question 14 Explanation:
Topic: Analyze and apply geometric transformations (e.g., translations, rotations, reflections, dilations); relate them to concepts of symmetry, similarity, and congruence; and use these concepts to solve problems (Objective 0024).
There are 14 questions to complete.

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