15 extra multiple-choice questions for Geometry (10th Standard Mathematics, Samacheer Kalvi), beyond the ones printed in the textbook — each with the correct option highlighted and a clear, worked explanation. Free to read in English and Tamil.
Q1
If the scale factor of two similar triangles is \(3 : 4\), what is the ratio of their corresponding areas?
- A. \(3 : 4\)
- B. \(9 : 16\)Correct
- C. \(16 : 9\)
- D. \(27 : 64\)
Explanation. According to the properties of similar triangles, the ratio of the areas of two similar triangles is equal to the square of their scale factor. Therefore, the ratio of their areas is \(3^2 : 4^2 = 9 : 16\).
Q2
In \(\Delta ABC\), a line \(DE\) is drawn parallel to \(BC\) intersecting \(AB\) at \(D\) and \(AC\) at \(E\). If \(AD = 4\) cm, \(DB = 6\) cm, and \(AE = 6\) cm, what is the length of \(EC\)?
- A. 4 cm
- B. 9 cmCorrect
- C. 10 cm
- D. 12 cm
Explanation. By Thales Theorem, the line parallel to one side of a triangle divides the other two sides proportionally. Therefore, \(\frac{AD}{DB} = \frac{AE}{EC}\), which gives \(\frac{4}{6} = \frac{6}{EC}\), resulting in \(EC = 9\) cm.
Q3
In \(\Delta PQR\), \(PS\) is the internal bisector of \(\angle P\) meeting \(QR\) at \(S\). If \(PQ = 12\) cm, \(PR = 15\) cm, and \(QS = 4\) cm, find the length of \(SR\).
- A. 5 cmCorrect
- B. 6 cm
- C. 8 cm
- D. 10 cm
Explanation. By the Angle Bisector Theorem, the internal bisector of an angle divides the opposite side internally in the ratio of the sides containing the angle. Thus, \(\frac{PQ}{PR} = \frac{QS}{SR} \implies \frac{12}{15} = \frac{4}{SR} \implies SR = 5\) cm.
Q4
A ladder of length 13 m is leaned against a vertical wall. If the foot of the ladder is 5 m away from the base of the wall, at what height does it reach the wall?
- A. 8 m
- B. 12 mCorrect
- C. 14 m
- D. 15 m
Explanation. Applying Pythagoras theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Thus, \(13^2 = 5^2 + h^2 \implies 169 = 25 + h^2 \implies h = 12\) m.
Q5
A tangent \(PA\) is drawn from an external point \(P\) to a circle of radius 6 cm with centre \(O\). If the length of the tangent \(PA\) is 8 cm, find the distance \(OP\).
- A. 10 cmCorrect
- B. 12 cm
- C. 14 cm
- D. 16 cm
Explanation. A tangent to a circle is perpendicular to the radius at the point of contact. In right-angled \(\Delta OAP\), \(OP^2 = OA^2 + PA^2\). Substituting the values gives \(OP^2 = 6^2 + 8^2 = 100 \implies OP = 10\) cm.
Q6
If two tangents \(PA\) and \(PB\) are drawn to a circle with centre \(O\) from an external point \(P\) such that \(\angle APB = 80^{\circ}\), then what is the measure of \(\angle AOP\)?
- A. \(40^{\circ}\)
- B. \(50^{\circ}\)Correct
- C. \(60^{\circ}\)
- D. \(70^{\circ}\)
Explanation. The line \(OP\) bisects \(\angle APB\), so \(\angle APO = 40^{\circ}\). Since the radius \(OA\) is perpendicular to tangent \(PA\), in right-angled \(\Delta OAP\), \(\angle AOP = 90^{\circ} - 40^{\circ} = 50^{\circ}\).
Q7
In \(\Delta ABC\), the cevians \(AD\), \(BE\), and \(CF\) are concurrent. If \(BD = 3\) cm, \(DC = 4\) cm, \(CE = 4\) cm, \(EA = 5\) cm, and \(AF = 10\) cm, what is the length of \(FB\)?
- A. 5 cm
- B. 6 cmCorrect
- C. 7.5 cm
- D. 8 cm
Explanation. According to Ceva's Theorem, for concurrent cevians, \(\frac{BD}{DC} \times \frac{CE}{EA} \times \frac{AF}{FB} = 1\). Substituting the values, we get \(\frac{3}{4} \times \frac{4}{5} \times \frac{10}{FB} = 1\), which simplifies to \(FB = 6\) cm.
Q8
A line cuts the sides \(BC\), \(CA\), and \(AB\) (or their extensions) of \(\Delta ABC\) at points \(D\), \(E\), and \(F\) respectively. If \(BD = 6\) cm, \(DC = 3\) cm, \(CE = 2\) cm, \(EA = 4\) cm, and \(AF = 3\) cm, find the length of \(FB\)?
- A. 1.5 cm
- B. 3 cmCorrect
- C. 4.5 cm
- D. 6 cm
Explanation. By Menelaus's Theorem, the collinear points on the sides of a triangle satisfy \(\frac{BD}{DC} \times \frac{CE}{EA} \times \frac{AF}{FB} = 1\). Substituting the values, we obtain \(\frac{6}{3} \times \frac{2}{4} \times \frac{3}{FB} = 1 \implies FB = 3\) cm.
Q9
In a circle with centre \(O\), two tangents \(PA\) and \(PB\) are drawn from an external point \(P\). If the angle between the two tangents is \(110^{\circ}\), what is the measure of the central angle \(\angle AOB\)?
- A. \(70^{\circ}\)Correct
- B. \(80^{\circ}\)
- C. \(90^{\circ}\)
- D. \(110^{\circ}\)
Explanation. The angle between two tangents from an external point and the angle subtended by the radii at the centre are supplementary. Therefore, the measure of \(\angle AOB\) is equal to \(180^{\circ} - 110^{\circ} = 70^{\circ}\).
Q10
In a circle, a chord \(AB\) makes an angle of \(65^{\circ}\) with the tangent \(BT\) at the point of contact \(B\). What is the angle \(\angle APB\) subtended by the chord in the major alternate segment?
- A. \(65^{\circ}\)Correct
- B. \(115^{\circ}\)
- C. \(130^{\circ}\)
- D. \(50^{\circ}\)
Explanation. According to the Alternate Segment Theorem, the angle between a tangent and a chord through the point of contact is equal to the angle subtended by the chord in the alternate segment. Thus, \(\angle APB = 65^{\circ}\).
Q11
If the perimeters of two similar triangles \(\Delta ABC\) and \(\Delta PQR\) are 45 cm and 30 cm respectively, and \(AB = 15\) cm, what is the length of the corresponding side \(PQ\)?
- A. 10 cmCorrect
- B. 12 cm
- C. 8 cm
- D. 15 cm
Explanation. For similar triangles, the ratio of their perimeters is equal to the ratio of their corresponding sides. Thus, \(\frac{\text{Perimeter}(\Delta ABC)}{\text{Perimeter}(\Delta PQR)} = \frac{AB}{PQ} \implies \frac{45}{30} = \frac{15}{PQ} \implies PQ = 10\) cm.
Q12
In \(\Delta ABC\), \(AB = 6\) cm, \(AC = 8\) cm, and \(AD\) is a line segment from \(A\) to \(BC\) such that \(BD = 3\) cm and \(DC = 4\) cm. Which of the following is true?
- A. \(AD\) is parallel to \(BC\)
- B. \(AD\) is perpendicular to \(BC\)
- C. \(AD\) is the angle bisector of \(\angle BAC\)Correct
- D. \(AD\) is the median of \(BC\)
Explanation. Since \(\frac{AB}{AC} = \frac{6}{8} = \frac{3}{4}\) and \(\frac{BD}{DC} = \frac{3}{4}\), we have \(\frac{AB}{AC} = \frac{BD}{DC}\). By the converse of the Angle Bisector Theorem, \(AD\) must be the internal angle bisector of \(\angle BAC\).
Q13
How many tangents can be drawn to a circle from a point lying strictly inside the circle?
- A. ZeroCorrect
- B. One
- C. Two
- D. Infinitely many
Explanation. A tangent is a line that touches a circle at exactly one point. Any line drawn through an interior point will intersect the circle at two distinct points, making it a secant line rather than a tangent.
Q14
If the sides of a right-angled triangle are \(x - 1\), \(x\), and \(x + 1\) (where \(x > 1\)), what is the value of \(x\)?
- A. 3
- B. 4Correct
- C. 5
- D. 6
Explanation. Using Pythagoras theorem, the square of the longest side is the sum of the squares of the other sides: \((x + 1)^2 = (x - 1)^2 + x^2 \implies x^2 + 2x + 1 = x^2 - 2x + 1 + x^2\), which simplifies to \(x^2 - 4x = 0 \implies x = 4\).
Q15
Two concentric circles have radii 13 cm and 5 cm. A chord of the larger circle is tangent to the smaller circle. Find the length of this chord.
- A. 12 cm
- B. 18 cm
- C. 24 cmCorrect
- D. 26 cm
Explanation. The radius to the point of contact is perpendicular to the tangent chord and bisects it. In the right-angled triangle formed, the half-chord length \(y\) satisfies \(13^2 = 5^2 + y^2 \implies y = 12\) cm. Thus, the chord length is \(24\) cm.
