Secants, Tangents And Their Properties
NIOS · Class 10 · Maths
Most important questions from Secants, Tangents And Their Properties for NIOS Class 10 Maths board exam 2026. MCQs, short answer, and long answer questions with marks.
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A secant PAB and tangent PT are drawn from external point P to a circle. If PA = 8 cm, AB = 10 cm, then PT equals:
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12 cm
Step 1: Given - PA = 8 cm, AB = 10 cm, so PB = PA + AB = 18 cm. Step 2: Apply the secant-tangent theorem: PA × PB = PT². Step 3: Substitute: 8 × 18 = PT². Step 4: 144 = PT². Step 5: PT = √144 = 12 cm. This theorem relates the secant segments to the tangent length from the same external point.
In triangle ABC, the incircle touches sides BC, CA, and AB at points X, Y, and Z respectively. If AZ = 4 cm, BX = 5 cm, and CY = 3 cm, the perimeter of triangle ABC is:
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24 cm
Step 1: Given - AZ = 4 cm, BX = 5 cm, CY = 3 cm. Step 2: Apply tangent property: tangents from external point are equal. So AY = AZ = 4 cm, BZ = BX = 5 cm, CX = CY = 3 cm. Step 3: Calculate sides: AB = AZ + ZB = 4 + 5 = 9 cm, BC = BX + XC = 5 + 3 = 8 cm, CA = CY + YA = 3 + 4 = 7 cm. Step 4: Perimeter = AB + BC + CA = 9 + 8 + 7 = 24 cm.
Two chords of a circle intersect outside the circle. If the segments are 6 cm, 9 cm on one chord and 4 cm, x cm on the other chord, then x equals:
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13.5 cm
Step 1: For chords intersecting outside, let segments be PA = 6 cm, PB = 6 + 9 = 15 cm, PC = 4 cm, PD = 4 + x cm. Step 2: Apply theorem: PA × PB = PC × PD. Step 3: 6 × 15 = 4 × (4 + x). Step 4: 90 = 16 + 4x. Step 5: 4x = 74, so x = 18.5. Wait, let me recalculate: if segments are 6, 9 and 4, x, then 6 × 9 = 4 × x, so x = 54/4 = 13.5 cm.
XY is a tangent to a circle at point P. If chord PQ makes an angle of 40° with tangent XY, then the angle inscribed in the alternate segment is:
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40°
Step 1: Given - XY is tangent at P, chord PQ makes 40° with tangent. Step 2: Apply the alternate segment theorem. Step 3: The angle between chord and tangent equals the angle inscribed in the alternate segment. Step 4: Therefore, the inscribed angle in alternate segment = 40°. Step 5: This is a direct application of the tangent-chord angle theorem.
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