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100+ Free Madrid PAU Technical Drawing Applied to Arts and Design II Practice Questions
Madrid PAU Technical Drawing Applied to Arts and Design II (Dibujo Técnico Aplicado a las Artes Plásticas y al Diseño II - UCM 2026) practice questions are available now; exam metadata is being verified.
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2026 Statistics
Key Facts: Madrid PAU Technical Drawing Applied to Arts and Design II Exam
90 Min
Official examination time limit
Madrid PAU Organising Commission UCM 2026
0–10
Official grading scale for PAU subject papers
Spanish University Access Regulations
Min 4.0
Minimum mark required in Access Phase to compute composite score
Community of Madrid University Admissions
EUR 93.02
Base registration fee for compulsory Access Phase in Madrid
Public University Fees Decree Madrid
100
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Madrid PAU Technical Drawing Applied to Arts and Design II tests students on plane geometry, tangencies, dihedral system projections, axonometric and conic perspective, and ISO normalization over 90 minutes.
Sample Madrid PAU Technical Drawing Applied to Arts and Design II Practice Questions
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1In planar triangle geometry, which notable point is defined as the common intersection of the three perpendicular bisectors (mediatrices) of the triangle's sides?
A.Circumcenter (Circuncentro)
B.Incenter (Incentro)
C.Orthocenter (Ortocentro)
D.Centroid (Baricentro)
Explanation: The circumcenter is the point where the three perpendicular bisectors of a triangle intersect. It serves as the center of the circumscribed circle passing through all three vertices. The incenter is formed by angle bisectors, the orthocenter by altitudes, and the centroid by medians.
2Point P lies at a distance d = 13 cm from the center O of a circle with radius R = 5 cm. What is the power of point P with respect to this circle?
A.64 cm²
B.144 cm²
C.169 cm²
D.25 cm²
Explanation: The power of a point P with respect to a circle of radius R centered at O is calculated as P = d² - R². Substituting d = 13 cm and R = 5 cm gives P = 13² - 5² = 169 - 25 = 144 cm². This value also equals the square of the length of the tangent segment from P to the circle.
3What is the defining geometric locus property of the radical axis (eje radical) of two non-concentric planar circles?
A.The line connecting the centers of both circles
B.The set of points from which both circles subtend equal visual angles
C.The straight line containing all points that have equal power with respect to both circles
D.The line perpendicular to the segment connecting the two centers passing through the midpoint
Explanation: The radical axis of two non-concentric circles is defined as the locus of all points in the plane that have equal power with respect to both circles. It is always a straight line perpendicular to the line of centers.
4In a planar inversion transformation with center O and positive constant of inversion K = 36 cm², point A is located 4 cm from O on a ray. What is the distance from O to the inverse point A'?
A.32 cm
B.144 cm
C.4.5 cm
D.9.0 cm
Explanation: By definition of planar inversion, points A and A' lie on the same ray from O such that OA · OA' = K. Substituting OA = 4 cm and K = 36 cm² yields OA' = 36 / 4 = 9.0 cm.
5Which fundamental geometric condition determines the line tangent to a circle at a specified point T on its circumference?
A.The tangent line is perpendicular to the radius drawn to the point of contact T
B.The tangent line forms a 45-degree angle with the radius at T
C.The tangent line passes through the center of the circle
D.The tangent line is parallel to the diameter passing through T
Explanation: A line is tangent to a circle at point T if and only if it is perpendicular (forms a 90-degree angle) to the radius OT connecting the center to the contact point T.
6To construct the two tangent lines from an exterior point P to a circle centered at O, which classic geometric construction is used to locate the tangent points T1 and T2?
A.Draw an equilateral triangle on segment OP
B.Draw Thales' circle using segment OP as diameter; its intersections with the circle yield T1 and T2
C.Inscribe a square inside the original circle and project its vertices to P
D.Bisect the radius of the circle and draw a parallel line through P
Explanation: Thales' circle theorem states that an angle inscribed in a semicircle is a right angle. Drawing an auxiliary circle with diameter OP intersects the original circle at T1 and T2, where angle OPT1 and OPT2 are 90 degrees, defining the exact points of tangency.
7Three non-concentric circles have non-collinear centers. How is their unique radical center (centro radical) constructed?
A.By finding the centroid of the triangle formed by their three centers
B.By constructing the circumcenter of the triangle formed by their radii
C.By determining the mutual intersection point of the three pairwise radical axes of the circles
D.By drawing the common external tangents and finding their midpoint
Explanation: The radical center of three circles with non-collinear centers is the single point having equal power with respect to all three circles. It is found at the intersection of the three pairwise radical axes.
8What is the exact algebraic expression for the golden ratio φ (número de oro / razón áurea)?
A.(1 + √3) / 2 ≈ 1.366
B.(1 + √2) / 2 ≈ 1.207
C.√5 - 1 ≈ 1.236
D.(1 + √5) / 2 ≈ 1.618
Explanation: The golden ratio φ is defined by the positive solution to the quadratic equation φ² - φ - 1 = 0, which yields φ = (1 + √5) / 2 ≈ 1.618033988...
9In ruler-and-compass geometry, a regular pentagon is inscribed in a circle of radius R. What is the length of its side s in terms of R?
A.s = (R / 2) · √(10 - 2√5) ≈ 1.176 R
B.s = R · √3 ≈ 1.732 R
C.s = R · √2 ≈ 1.414 R
D.s = R · (√5 - 1) ≈ 1.236 R
Explanation: The side of an inscribed regular pentagon in a circle of radius R is s = 2R · sin(36°) = (R / 2) · √(10 - 2√5) ≈ 1.1756 R. R·√3 is the side of an inscribed equilateral triangle, and R·√2 is the side of an inscribed square.
10In a planar inversion transformation with center O, what is the inverse image of a straight line r that passes directly through the center of inversion O?
A.A circle passing through O
B.The line r itself (invariant as a set of points)
C.A parabola with vertex at O
D.A circle not passing through O
Explanation: Any straight line passing through the center of inversion O is transformed into itself (the line is invariant as a whole entity, though individual points exchange positions along rays from O).
About the Madrid PAU Technical Drawing Applied to Arts and Design II Practice Questions
Verified exam format metadata for Madrid PAU Technical Drawing Applied to Arts and Design II (Dibujo Técnico Aplicado a las Artes Plásticas y al Diseño II - UCM 2026) is pending. The practice questions above remain available while official exam length, timing, passing score, fee, and administrator details are reviewed.