## Basic Geometry Practice

**Note: Figure not drawn to scale**

** 1. Calculate the length of side x.**

a. 6.46

b. 8.48

c. 3.6

d. 6.4

**Note: figure not drawn to scale**

** 2. What is the length of each side of the indicated square**

** above? Assume the 3 shapes around the center triangle**

** are square.**

a. 10

b. 15

c. 20

d. 5

**3. Reflect the parallelogram ABCD with the given mirror**

** line m.**

**4. In a class, there are 8 students who take only dancing ****lessons. 5 students take both dancing and singing lessons. ****The number of students taking singing lessons is 1 ****less than 2 times the number of students taking dancing ****lessons only. If 4 students take neither of these courses,**

** how many students are in the class?**

a. 19

b. 25

c. 27

d. 42

**5. The interior angles of a triangle are given as 2x + 5, 6x**

** and 3x – 23. Find the supplementary of the largest angle.**

a. 64^{0}

b. 72^{0}

c. 100^{0}

d. 108^{0}

**6. In the right triangle above, |AB| = 2|BC| and |AC| =**

** 15 cm. Find the length of |AB|.**

a. 5 cm

b. 10 cm

c. 5√5 cm

d. 6√5 cm

**Note: figure not drawn to scale**

**7. What is the perimeter of the equilateral ΔABC above?**

a. 18 cm

b. 12 cm

c. 27 cm

d. 15 cm

**Note: figure not drawn to scale**

**8. What is the perimeter of the above shape, assuming**

** the bottom portion is square?**

a. 22.85 cm

b. 20 π cm

c. 15 π cm

d. 25 π cm

**9. What year was Euclidian geometry disproven, and by**

** whom?**

a. Thales – BC 500s

b. Pythagor BC 500s

c. Pierre De Fermat 1600s

d. Nikolai Lobachevsky 1830s

**10. Which postulate below disproves Euclidean**

** geometry?**

a. Through any two points, there is exactly one line.

b. If equals are added to equals, the wholes are equal.

c. Parallel postulate

d. Things which coincide with one another are equal to

one another (Reflexive property).

**Answer Key **

**1. B**

In the question, we have a right triangle formed inside the circle. We are asked to find the length of the hypotenuse of this triangle. We can find the other two sides of the triangle by using circle properties:

The diameter of the circle is equal to 12 cm. The legs of the

right triangle are the radii of the circle; so they are 6 cm

long.

Using the Pythagorean Theorem:

(Hypotenuse)^{2} = (Adjacent Side)^{2} + (Opposite Side)^{2}

x^{2} = r^{2} + r^{2}

x^{2} = 6^{2} + 6^{2}

x^{2} = 72

x = √72

x = 8.48

**2. B**

We see that there are three squares forming a right triangle in the middle. Two of the squares have the areas 81 m^{2} and 144 m^{2}. If we denote their sides a and b respectively:

a^{2} = 81 and b^{2} = 144. The length, which is asked, is the hypotenuse; a and b are the opposite and adjacent sides of the right angle. By using the Pythagorean Theorem, we can find the value of the asked side:

Pythagorean Theorem:

(Hypotenuse)^{2} = (Opposite Side)^{2} + (Adjacent Side)^{2}

h^{2} = a^{2} + b^{2}

a^{2} = 81 and b^{2} = 144 are given. So,

h^{2} = 81 + 144

h^{2} = 225

h = 15 m

**3.**

We reflect points A, B, C and D against the mirror line m at

right angle and we connect the new points A’, B’,C’ and D’.

**4. C**

The class can be shown by set E. Let us say, set D represents dancing lessons, and set S represents singing lessons. 8 students take only dancing lessons: s(DS) = 8

5 students take both dancing and singing lessons: s(D∩S) = 5

The number of students taking singing lessons:

s(S) = 2 * 8 – 1 = 15

s(S) = s(SD) + s(D∩S) and also s(D) = s(DS) + s(D∩S)

4 students take neither of these courses: s(E) – s(D ∪ S) = 4

s(D ∪ S) can be found by s(DS) + s(S) or s(SD) + s(D)

We are asked to find s(E):

s(E) = 4 + s(DS) + s(S) = 4 + 8 + 15 = 27

**5. B**

The interior angles of a triangle sum up to 1800:

(2x + 5) + (6x) + (3x – 23) = 180

82 NYSTCE® Mathematics Skill Practice!

11x – 18 = 180

11x = 198

x = 180

The largest angle is 6x = 6 * 18 = 1080

The supplementary of an angle is the angle which plus the

angle gives 1800. Then, the supplementary of 1080 is:

180 – 108 = 720

**6. D**

In the right triangle above, AB and BC are the legs and AC is the hypotenuse. For side lengths, Pythagorean Theorem is applied:

|AB|^{2} + |BC|^{2} = |AC|^{2}

Let us say that |BC| = x. Then, |AB| = 2x:

(2x)^{2} + x^{2} = 152

5x^{2} = 225

x^{2} = 45

x = √45 = 3√5 cm

|AB| = 2x → |AB| = 6√5 cm

**7. C**

The perimeter of an equilateral triangle with 9 cm. sides will be

= 9 + 9 + 9 = 27 cm.

**8. A**

The question is to find the perimeter of a shape made by merging a square and a semi circle. Perimeter = 3 sides of the square + ½ circumference of the circle.

= (3 x 5) + ½(5 π)

= 15 + 2.5 π

Perimeter = 22.85 cm

**9. D**

Euclidian geometry was disproven by Nikolai Lobachevsky in 1830s.

**10. C**

Euclidian geometry supports parallel geometry. On the contrary, Non-Euclidian geometry is the study of geometry with curved spaces that is elliptic and hyperbolic geometry. In elliptic geometry; the inner angles of a triangle do not sum up to 1800; the sum is equal to 1800 plus the area of

the triangle. In hyperbolic geometry; the sum is equal to 1800 minus the area of the triangle. non-Euclidian geometry inspires from the shape of the world: If two meridians are selected; both intersect with the equator by 900. There is also a vertex angle in the pole. So, the inner angles sum up to 90 + 90 + vertex pole which is higher than 1800. Another example; a person walks 10 m south, 10 m west and then 10 m north. He sees that he is where he started moving. Normally, we would say that he would be 10 m east from the starting point. Think that he is on the North Pole.

If he goes 10 m down, 10 m west and then 10 m north, he is

again on the pole.

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