Puerto Rico CRECE Mathematics — Grade 10

What is the distance between points $(1, 2)$ and $(4, 6)$ in the coordinate plane?

• A5
• B7
• C\sqrt{7}
• D25

What is the midpoint of the segment connecting points $(-2, 8)$ and $(6, 2)$?

• A(2, 5)
• B(4, 10)
• C(4, 3)
• D(2, 3)

What is the radius of a circle defined by the equation $(x - 3)^2 + (y + 2)^2 = 49$?

• A7
• B14
• C49
• D\sqrt{7}

In a 45°-45°-90° right triangle, each leg has a length of 6 cm. What is the exact length of the hypotenuse?

• A6
• B6\sqrt{2}
• C6\sqrt{3}
• D12

In a 30°-60°-90° right triangle, the hypotenuse measures 12 cm. What is the length of the side opposite the 60° angle?

• A6 cm
• B6\sqrt{3} cm
• C12\sqrt{3} cm
• D6\sqrt{2} cm

What is the equation of the line perpendicular to $y = -\frac{1}{2}x + 4$ that passes through point $(3, 1)$?

• Ay = 2x - 5
• By = 2x + 7
• Cy = -\frac{1}{2}x + \frac{5}{2}
• Dy = -2x + 7

What are the center coordinates and radius length of the circle with equation $x^2 + y^2 - 6x + 8y = 0$?

• ACenter (3, -4), radius = 5
• BCenter (-3, 4), radius = 5
• CCenter (3, -4), radius = 25
• DCenter (6, -8), radius = 10

Triangle ABC has vertices $A(0, 0)$, $B(4, 0)$, and $C(2, 2\sqrt{3})$. What type of triangle is ABC?

• AIsosceles right triangle
• BScalene acute triangle
• CEquilateral triangle
• DObtuse triangle

What is the standard equation of the circle passing through the points $(0, 0)$, $(6, 0)$, and $(0, 8)$?

• A(x - 3)^2 + (y - 4)^2 = 25
• B(x - 6)^2 + (y - 8)^2 = 100
• C(x - 3)^2 + (y - 4)^2 = 5
• Dx^2 + y^2 = 25

What is the equation of the perpendicular bisector of the line segment joining points $A(2, 5)$ and $B(8, 1)$?

• A3x - 2y = 9
• B2x + 3y = 19
• C3x + 2y = 21
• D2x - 3y = 1

In Euclidean geometry, which statement about a plane is accepted as true?

• AThrough any three points not on the same line there is exactly one plane
• BThrough any three points not on the same line there are infinitely many planes
• CA plane has a definite length and width but no thickness only if it is shaded
• DTwo distinct planes can intersect in exactly one point

Which figure has exactly one endpoint and extends without end in one direction?

• AA ray
• BA line segment
• CA line
• DA plane

Two lines intersect, forming four angles. One of the angles measures 118°. What is the measure of the angle vertical to it?

• A118°
• B62°
• C42°
• D236°

In triangle ABC, m∠A = 53° and m∠B = 74°. What is m∠C?

• A53°
• B127°
• C63°
• D43°

Which property is true of every parallelogram?

• AOpposite sides are congruent and opposite angles are congruent
• BAll four sides are congruent
• CThe diagonals are always congruent
• DAll four interior angles are right angles

Two parallel lines are cut by a transversal. Which pair of angles must be supplementary?

• AA pair of co-interior angles on the same side of the transversal
• BA pair of corresponding angles
• CA pair of alternate interior angles
• DA pair of alternate exterior angles

An exterior angle of a triangle measures 128°. One of its remote interior angles measures 47°. What is the measure of the other remote interior angle?

• A81°
• B52°
• C175°
• D43°

In isosceles triangle PQR, PQ ≅ PR and m∠P = 46°. What is the measure of ∠Q?

• A67°
• B46°
• C88°
• D134°

In parallelogram ABCD, m∠A = 68°. What are the measures of ∠B and ∠C?

• Am∠B = 112° and m∠C = 68°
• Bm∠B = 68° and m∠C = 112°
• Cm∠B = 112° and m∠C = 112°
• Dm∠B = 22° and m∠C = 68°

The midsegment of a triangle joins the midpoints of two sides. Which statement about the midsegment is true?

• AIt is parallel to the third side and half as long
• BIt is perpendicular to the third side and half as long
• CIt is parallel to the third side and equal in length to it
• DIt always passes through the centroid and bisects the third side

A line is drawn parallel to side BC of triangle ABC, meeting AB at D and AC at E. If AD = 6, DB = 4, and AE = 9, what is EC?

• A6
• B4
• C13.5
• D7.5

Triangle ABC has vertices A(1, 2), B(5, 2), and C(5, 8). What is the length of the hypotenuse AC?

• AAbout 7.2 units
• B10 units
• CAbout 5.7 units
• D6 units

Which set of side lengths cannot form a triangle?

• A5, 7, 13
• B5, 7, 11
• C6, 8, 10
• D9, 9, 9

Point M lies on segment AB such that AM ≅ MB, and line ℓ passes through M perpendicular to AB. Which statement about line ℓ is guaranteed?

• AEvery point on ℓ is equidistant from A and B
• BEvery point on ℓ is closer to A than to B
• CLine ℓ must pass through the midpoint of every segment in the plane
• DLine ℓ is parallel to AB

Which construction can be performed with only a compass and an unmarked straightedge?

• ABisecting a given angle
• BMeasuring an angle in degrees
• CConstructing a segment of exactly 7.3 centimetres without a ruler
• DTrisecting an arbitrary angle

Which sequence of compass-and-straightedge steps constructs an equilateral triangle on a given segment AB?

• ADraw arcs of radius AB centred at A and at B, then join an intersection point of the arcs…
• BBisect AB, then draw a perpendicular at the midpoint and connect its endpoint to A
• CDraw a circle through A and B with any radius, then connect A and B to the circle's centre
• DCopy angle A onto B and extend both rays until they meet

A geometry student writes: "If a quadrilateral is a rectangle, then its diagonals are congruent." What is the converse of this statement, and is the converse true for all quadrilaterals?

• AConverse: if a quadrilateral's diagonals are congruent, then it is a rectangle; this conv…
• BConverse: if a quadrilateral is not a rectangle, then its diagonals are not congruent; th…
• CConverse: if a quadrilateral's diagonals are congruent, then it is a rectangle; this conv…
• DConverse: if a quadrilateral has congruent diagonals, then it is a rhombus; this converse…

In a two-column proof, a student concludes that ∠1 ≅ ∠3 after establishing that ∠1 and ∠2 form a linear pair, ∠2 and ∠3 form a linear pair, and both linear pairs share ∠2. Which justification completes the proof?

• AAngles supplementary to the same angle are congruent
• BVertical angles are congruent
• CAll right angles are congruent
• DCorresponding angles of parallel lines are congruent

In a coordinate proof, quadrilateral WXYZ has vertices W(0, 0), X(6, 0), Y(8, 4), and Z(2, 4). Which conclusion is best supported by calculating slopes and side lengths?

• AWXYZ is a parallelogram, because both pairs of opposite sides are parallel and congruent,…
• BWXYZ is a rectangle, because all four angles are right angles
• CWXYZ is a trapezoid with exactly one pair of parallel sides
• DWXYZ is a rhombus, because all four sides are congruent

Two parallel lines are cut by a transversal so that one pair of alternate interior angles measures (5x − 12)° and (3x + 20)°. What is the measure of each of these angles?

• A68°
• B16°
• C56°
• D80°

A student claims that any two triangles with two pairs of congruent sides and one pair of congruent angles must be congruent. Which observation best refutes this claim?

• AThe congruent angle must be the included angle between the two sides; otherwise a side-si…
• BTwo pairs of congruent sides are never enough information, because at least three pairs o…
• CThe claim fails because congruent triangles must also have equal areas, which cannot be c…
• DThe claim is actually correct, since side-side-angle is a valid congruence criterion

Which transformation is NOT a rigid motion?

• AA dilation with scale factor 3 centred at the origin
• BA translation 4 units left and 2 units up
• CA reflection across the line y = x
• DA rotation of 90° about the origin

Point P(3, −5) is rotated 180° about the origin. What are the coordinates of its image P′?

• A(−3, 5)
• B(3, 5)
• C(−3, −5)
• D(5, −3)

Why are all circles similar to one another?

• AAny circle can be mapped onto any other by a translation followed by a dilation
• BAll circles have the same radius by definition
• CAll circles have the same area regardless of radius
• DCircles cannot be compared, so similarity does not apply to them

A radius of a circle is drawn to the point where a tangent line touches the circle. What is the measure of the angle between the radius and the tangent?

• A90°
• B45°
• C180°
• DIt varies depending on the radius

Triangle ABC is mapped onto triangle DEF by a reflection across the y-axis followed by a translation. What can be concluded about the two triangles?

• AThey are congruent, because the mapping is a composition of rigid motions
• BThey are similar but not congruent, because a reflection changes size
• CThey are neither similar nor congruent, because two transformations were applied
• DThey are congruent only if the triangles are equilateral

Triangle ABC has vertices A(2, 1), B(4, 1), and C(2, 5). A dilation centred at the origin with scale factor 3 maps it to triangle A′B′C′. What are the coordinates of C′?

• A(6, 15)
• B(5, 8)
• C(2, 15)
• D(6, 5)

Two triangles have two pairs of congruent angles. Which conclusion is justified?

• AThe triangles are similar by the AA criterion, and their corresponding sides are proporti…
• BThe triangles are congruent, because two angles determine a triangle completely
• CNothing can be concluded without knowing at least one side length
• DThe triangles are similar only if the third angles are also measured

Triangle RST is similar to triangle XYZ with RS = 8, ST = 12, and XY = 6. What is the length of YZ?

• A9
• B10
• C16
• D4

A rectangular photograph 12 cm by 18 cm is enlarged by a scale factor of 2.5. How does the area of the enlargement compare with the area of the original?

• AIt is 6.25 times as large, because area scales by the square of the linear scale factor
• BIt is 2.5 times as large, because every length was multiplied by 2.5
• CIt is 5 times as large, because both dimensions were doubled and then halved
• DIt is 15.625 times as large, because area scales by the cube of the scale factor

In a circle, a central angle measures 70°. What is the measure of an inscribed angle that intercepts the same arc?

• A35°
• B70°
• C140°
• D110°

Triangle ABC is inscribed in a circle so that side AC is a diameter. What must be true of angle B?

• AIt is a right angle, because an angle inscribed in a semicircle measures 90°
• BIt measures 45°, because a diameter divides the circle into two equal parts
• CIt measures 180°, because AC passes through the centre
• DIts measure depends on the length of the diameter

Quadrilateral PQRS is inscribed in a circle, and m∠P = 84°. What is m∠R?

• A96°
• B84°
• C42°
• D168°

Which point of concurrency is the centre of the circle that passes through all three vertices of a triangle?

• AThe circumcenter, found where the perpendicular bisectors of the sides meet
• BThe incenter, found where the angle bisectors meet
• CThe centroid, found where the medians meet
• DThe orthocenter, found where the altitudes meet

A regular hexagon is inscribed in a circle of radius 9 cm. What is the length of each side of the hexagon?

• A9 cm
• B4.5 cm
• C18 cm
• DAbout 7.8 cm

A municipality in Puerto Rico has an area of 60 square kilometres and a population of 45,000 residents. What is its population density?

• A750 residents per square kilometre
• B2,700,000 residents per square kilometre
• C0.00133 residents per square kilometre
• D45,060 residents per square kilometre

A design team must build a storage box with a square base and a volume of 32 cubic feet, using the least possible material. They compare a base of 4 ft by 4 ft with a height of 2 ft against a base of 2 ft by 2 ft with a height of 8 ft. Which comparison is correct?

• ABoth boxes hold 32 cubic feet, but the 4 ft by 4 ft by 2 ft box uses less surface area, 6…
• BBoth boxes hold 32 cubic feet and require exactly the same surface area
• CThe taller box holds more volume, so it is the better design
• DThe taller box uses less surface area, 64 square feet compared with 72 square feet