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2026 Statistics
Key Facts: Madrid PAU Technical Drawing II Exam
90 Mins
Official examination duration
Madrid PAU Regulations
Non-graphic calculators
Calculators without graphic display or symbolic capabilities are allowed
Comunidad de Madrid PAU Guidelines
EUR 93.02
Base registration fee for compulsory Access Phase in Madrid
Comunidad de Madrid Official Gazette
Min 4.0
Minimum Access Phase mark required to calculate university admission mark
Spanish University Admission Framework
100 Questions
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Sample Madrid PAU Technical Drawing II Practice Questions
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1In plane geometry, what defines a homothety (homotecia) with center O and ratio k = -1.5?
A.Every point P and its image P' lie on opposite sides of center O with OP' = 1.5 · OP.
B.Every point P and its image P' lie on the same ray from center O with OP' = 1.5 · OP.
C.The figure is rotated by 90 degrees around center O and scaled by a factor of 1.5.
D.The figure is translated along a vector of length 1.5 units from center O.
Explanation: A homothety with a negative ratio k = -1.5 places the transformed image point P' on the opposite side of the center of homothety O relative to P. The distance from the center satisfies OP' = |k| · OP = 1.5 · OP, resulting in an inverted image scaled by 150%.
2A point P is located 4 cm from the center of homothety O. Under a homothety of ratio k = -2.5, what is the distance between point P and its transformed image P'?
A.10 cm
B.14 cm
C.6 cm
D.16 cm
Explanation: Since k = -2.5, the image point P' lies on the opposite side of O relative to P. The distance from O to P' is OP' = |-2.5| · 4 cm = 10 cm. Because O lies between P and P', the total distance between P and P' is OP + OP' = 4 cm + 10 cm = 14 cm.
3In a plane inversion with center O, point P is at distance OP = 4 cm from O and its inverse point P' is at distance OP' = 9 cm. What is the radius R of the circle of inversion?
A.36 cm
B.6.5 cm
C.6 cm
D.5 cm
Explanation: The power of inversion K is given by K = OP · OP' = 4 cm · 9 cm = 36 cm². Since the radius of the circle of inversion R satisfies R² = K, we have R = √36 = 6 cm.
4What is the inverse of a circle that passes through the center of inversion O?
A.A concentric circle centered at the center of inversion O.
B.A circle passing through O of twice the original diameter.
C.A straight line passing directly through the center of inversion O.
D.A straight line not passing through O, perpendicular to the diameter passing through O.
Explanation: In plane inversion, any circle passing through the center of inversion O transforms into a straight line that does not pass through O. This line is perpendicular to the line connecting O and the center of the original circle.
5A point P is located at distance d = 10 cm from the center of circle C with radius R = 6 cm. What is the radical power of point P with respect to circle C?
A.64 cm²
B.16 cm²
C.100 cm²
D.36 cm²
Explanation: The power of a point P with respect to a circle centered at O with radius R is P(C) = d² - R², where d is the distance OP. Here, P(C) = 10² - 6² = 100 - 36 = 64 cm².
6What geometric locus forms the radical axis of two intersecting circles?
A.The line segment connecting the centers of the two circles.
B.The straight line passing through the two intersection points of the circles.
C.A circle concentric with the larger circle passing through the intersection points.
D.The line perpendicular to the line of centers passing through one of the centers.
Explanation: The radical axis of two circles is the locus of points having equal power with respect to both circles. For two intersecting circles, their intersection points both have zero power with respect to each circle, so the radical axis is the straight line passing through both intersection points.
7In a plane homology (homología), what is the key property of the axis of homology e?
A.It is perpendicular to all lines connecting corresponding points P and P'.
B.It is the line containing the center of homology O and all vanishing points.
C.It is the locus of invariant points; any line intersects its transformed image at a point on e.
D.It divides the plane into two regions with opposite homothety ratios.
Explanation: The axis of homology e consists entirely of double (invariant) points. A fundamental property of planar homology is that any line and its transformed image line always intersect at a point belonging to the axis of homology e.
8An affinity (afinidad) in the plane is a special case of homology. Which element of general homology is situated at infinity in an affinity?
A.The origin of coordinates.
B.The axis of homology e.
C.The vanishing line r_v.
D.The center of homology O.
Explanation: An affinity is a planar homology whose center of homology O is located at infinity. Consequently, lines connecting corresponding points P and P' are all parallel to a fixed direction known as the direction of affinity.
9A line segment AB of length L is divided into golden section (sección áurea) at point C such that AC is the major segment (AC > CB). What is the exact formula for the length of AC?
A.AC = L · (√5 - 1) / 2
B.AC = L · (√5 + 1) / 2
C.AC = L · (√3 - 1) / 2
D.AC = L · (√2 / 2)
Explanation: By definition of the golden ratio, L / AC = AC / CB, which yields the quadratic equation (AC)² + L · AC - L² = 0. Solving for AC gives AC = L · (√5 - 1) / 2 ≈ 0.61803 · L.
10Which three classic triangle centers always lie on the Euler line (recta de Euler) of any scalene triangle?
A.Incenter, Orthocenter, and Circumcenter.
B.Orthocenter, Centroid (Baricentro), and Circumcenter.
C.Incenter, Centroid, and Excenter.
D.Nine-point center, Incenter, and Centroid.
Explanation: The Euler line passes through the Orthocenter (H), Centroid (G), and Circumcenter (O) of any non-equilateral triangle. Furthermore, the centroid divides the segment between the orthocenter and circumcenter in a 2:1 ratio (HG = 2 · GO).
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