M3 Core Math Final Exam

Are these shapes similar?

• No, the two shapes are congruent but not similar

• Yes, the lengths of the sides are proportional

• Cannot be determined

Answer explanation

Yes, the shapes are similar because their corresponding side lengths are proportional, which is a key criterion for similarity in geometry.

What is the similarity statement for the similar triangles ⊿PRQ and ⊿XYZ?

• ⊿QPR~⊿XYZ

• ⊿QPR≠⊿XYZ

• ⊿QRP ∞ ⊿ZYX

Answer explanation

The correct similarity statement for the triangles ⊿PRQ and ⊿XYZ is ⊿QPR~⊿XYZ, indicating that the angles and sides of these triangles are proportional, confirming their similarity.

Solve for x:

• 5 cm

• 10 cm

• 15 cm

• 20 cm

Answer explanation

To solve for x, we can set up an equation based on the problem context. After solving, we find that x equals 15 cm, which is the correct answer among the choices provided.

Which equation is the Pythagorean Theorem?

• A=πr2

• x=-b±b2-4ac2a

• cosα+cosβ=2cos12α+βcos12α-β

Answer explanation

The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). Thus, the correct equation is a² + b² = c².

When do we use the Pythagorean Theorem?

• To find a missing side of a RIGHT triangle

• The sum of all angles in a triangle is 180°

• None of the above

Answer explanation

The Pythagorean Theorem is specifically used to find a missing side of a right triangle, relating the lengths of the sides through the equation a² + b² = c², where c is the hypotenuse.

Which symbol shows congruence?

• ≅

• ≠

• ∝

• ≪

Answer explanation

The symbol ≅ indicates congruence, meaning two figures have the same shape and size. The other symbols represent different concepts: ≠ means not equal, ∝ means proportional, and ≪ means much less than.

Which of the following is NOT a trigonometry ratio?

• Square

• Sine

• Cosine

• Tangent

Answer explanation

The correct answer is 'Square' because it is not a trigonometric ratio. The trigonometric ratios include Sine, Cosine, and Tangent, which relate the angles of a triangle to the lengths of its sides.

In a 45°-45°-90° triangle, the length of the hypotenuse is:

• a

• 2a

• None of the above

Answer explanation

In a 45°-45°-90° triangle, the hypotenuse is equal to the length of a leg multiplied by √2. If the legs are of length 'a', the hypotenuse is 'a√2', making the correct answer a√2.

In a 30°-60°-90° triangle, the length of the hypotenuse is:

• a

• b

• a3

• 2a

Answer explanation

In a 30°-60°-90° triangle, the hypotenuse is twice the length of the shorter leg (a). Therefore, if the shorter leg is 'a', the hypotenuse is '2a', making '2a' the correct answer.

What type of triangle ALWAYS has a 90° angle?

• Right

• Scalene

• Obtuse

• Acute

Answer explanation

A right triangle is defined as having one angle that measures exactly 90°. This is the distinguishing feature of right triangles, unlike scalene, obtuse, or acute triangles, which do not always have a 90° angle.