## 0024 Geometry

Hints will display for most wrong answers; explanations for most right answers. You can attempt a question multiple times; it will only be scored correct if you get it right the first time.

I used the official objectives and sample test to construct these questions, but cannot promise that they accurately reflect what’s on the real test. Some of the sample questions were more convoluted than I could bear to write. See terms of use. See the MTEL Practice Test main page to view random questions on a variety of topics or to download paper practice tests.

## MTEL General Curriculum Mathematics Practice

Question 1 |

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

\( \large 6x\) Hint: Use the Pythagorean Theorem. | |

\( \large 3x+5{{x}^{2}}\) Hint: Don't forget to take square roots when you use the Pythagorean Theorem. | |

\( \large 3x+\sqrt{5}{{x}^{2}}\) Hint: \(\sqrt {5 x^2}\) is not \(\sqrt {5}x^2\). | |

\( \large 3x+\sqrt{5}{{x}^{{}}}\) 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 2 |

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

\( \large 18+\sqrt{2} \text{ units}\) Hint: Be careful with the Pythagorean Theorem. | |

\( \large 18+2\sqrt{2}\text{ units}\) 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 \(2 \sqrt{2}\). | |

\( \large 18 \text{ units}
\) Hint: Use the Pythagorean Theorem to find the lengths of the diagonal segments. | |

\( \large 20 \text{ units}\) Hint: Use the Pythagorean Theorem to find the lengths of the diagonal segments. |

Question 3 |

#### What is the mathematical name of the three-dimensional polyhedron depicted below?

## TetrahedronHint: All the faces of a tetrahedron are triangles. | |

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

## Triangular PyramidHint: A pyramid has one base, not two. | |

## TrigonHint: A trigon is a triangle (this is not a common term). |

Question 4 |

#### Which of the following is not possible?

## 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 5 |

#### How many lines of reflective symmetry and how many centers of rotational symmetry does the parallelogram depicted below have?

## 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 6 |

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

Hint: If you have trouble visualizing, cut them out and fold (during the test, you can tear paper to approximate). | |

Hint: If you have trouble visualizing, cut them out and fold (during the test, you can tear paper to approximate). | |

Hint: If you have trouble visualizing, cut them out and fold (during the test, you can tear paper to approximate). |

Question 7 |

#### Which of the following sets of polygons can be assembled to form a pentagonal pyramid?

## 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 5 isosceles triangles. | |

## 1 pentagon and 10 isosceles triangles. |

Question 8 |

#### Which property is not shared by all rhombi?

## 4 congruent sidesHint: The most common definition of a rhombus is a quadrilateral with 4 congruent sides. | |

## A center of rotational symmetryHint: 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 anglesHint: Unless the rhombus is a square, it does not have 4 congruent angles. | |

## 2 sets of parallel sidesHint: All rhombi are parallelograms. |

Question 9 |

#### Below are front, side, and top views of a three-dimensional solid.

#### Which of the following could be the solid shown above?

## A sphereHint: All views would be circles. | |

## A cylinder | |

## A coneHint: Two views would be triangles, not rectangles. | |

## A pyramidHint: How would one view be a circle? |

Question 10 |

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

\( \large 20{}^\circ \) 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. | |

\( \large 25{}^\circ \) Hint: If two sides of a triangle are congruent, then it's isosceles, and the base angles of an isosceles triangle are equal. | |

\( \large 30{}^\circ \) Hint: If two sides of a triangle are congruent, then it's isosceles, and the base angles of an isosceles triangle are equal. | |

\( \large 40{}^\circ \) Hint: Make sure you're calculating the correct angle. |

Question 11 |

#### Use the graph below to answer the question that follows.

#### 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?

Hint: Try following the point (1,4) to see where it goes after each transformation. | |

Hint: Make sure you're reflecting in the correct axis. | |

Hint: Make sure you're rotating the correct direction. |

Question 12 |

#### Aya and Kendra want to estimate the height of a tree. On a sunny day, Aya measures Kendra’s shadow as 3 meters long, and Kendra measures the tree’s shadow as 15 meters long. Kendra is 1.5 meters tall. How tall is the tree?

## 7.5 metersHint: 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. | |

## 22.5 metersHint: Draw a picture. | |

## 30 metersHint: Draw a picture. | |

## 45 metersHint: Draw a picture. |

Question 13 |

#### What set of transformations will transform the leftmost image into the rightmost image?

## 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 (5,1), followed by a translation of 2 units up. | |

## 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 14 |

#### Use the four figures below to answer the question that follows:

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

\( \large 1\) | |

\( \large 2\) 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. | |

\( \large 3\) | |

\( \large 4\) Hint: All four have rotational symmetry, but not reflective symmetry. |

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