All questions
- In figure 3.97, circles with centres C and D touch internally at point E. D lies on the inner circle. Chord EB of the outer circle intersects inner circle at point A. Prove that, seg EA ≅ seg AB. Answer: ED is a diameter of the inner circle, so ∠ EAD = 90^° ; the perpendicular from centre D bisects chord EB. By the theorem of touching... 1
- In figure 3.98, seg AB is a diameter of a circle with centre O. The bisector of ∠ACB intersects the circle at point D. Prove that, seg AD ≅ seg BD. Complete the following proof by filling in the blanks. Answer: OD is the perpendicular bisector of AB, so D is equidistant from A and B. Draw seg OD. ∠ ACB = 90^° …… angle inscribed in semicircle... 1
- In figure 3.99, seg MN is a chord of a circle with centre O. MN = 25, L is a point on chord MN such that ML = 9 and d(O,L) = 5. Find the radius of the circle. Answer: The radius is 13. Draw OP ⊥ MN. The perpendicular from the centre bisects the chord, so MP = 25/2 = 12.5, and LP = MP − ML = 12.5 −... 1
- In figure 3.100, two circles intersect each other at points S and R. Their common tangent PQ touches the circle at points P, Q. Prove that, ∠PRQ + ∠PSQ = 180° Answer: By the tangent–secant angle theorem ∠ RPQ = ∠ PSR and ∠ RQP = ∠ QSR ; adding and using the angle sum of triangle PRQ gives the resul... 1
- ΔABC ~ ΔLMN. In ΔABC, AB = 5.5 cm, BC = 6 cm, CA = 4.5 cm. Construct ΔABC and ΔLMN such that BC/MN = 5/4. Answer: Sides of ΔLMN are LM = 4.4 cm, MN = 4.8 cm, LN = 3.6 cm. Analysis: ΔABC ~ ΔLMN, so AB/LM=BC/MN=CA/NL=5/4 . ΔLMN is the smaller trian... 1
- ΔPQR ~ ΔLTR. In ΔPQR, PQ = 4.2 cm, QR = 5.4 cm, PR = 4.8 cm. Construct ΔPQR and ΔLTR, such that PQ/LT = 3/4. Answer: Sides of ΔLTR are LT = 5.6 cm, TR = 7.2 cm, LR = 6.4 cm. Analysis: PQ/LT=QR/TR=PR/LR=3/4 , so ΔLTR is bigger. The vertex R is common... 1
- ΔRST ~ ΔXYZ. In ΔRST, RS = 4.5 cm, ∠RST = 40°, ST = 5.7 cm. Construct ΔRST and ΔXYZ, such that RS/XY = 3/5. Answer: In ΔXYZ, XY = 7.5 cm, ∠XYZ = 40°, YZ = 9.5 cm. Analysis: RS/XY=ST/YZ=3/5 and ∠ S = ∠ Y = 40^° . XY = 4.5×5/3 = 7.5 cm, YZ = 5.7×5/3... 1
- ΔAMT ~ ΔAHE. In ΔAMT, AM = 6.3 cm, ∠TAM = 50°, AT = 5.6 cm. AM/AH = 7/5. Construct ΔAHE. Answer: In ΔAHE, AH = 4.5 cm, ∠EAH = 50°, AE = 4 cm. Analysis: AM/AH=AT/AE=MT/HE=7/5 and ∠ TAM=∠ EAH=50^° . ΔAHE is smaller; A is common, so... 1
- Construct a tangent to a circle with centre P and radius 3.2 cm at any point M on it. Answer: Line MZ, perpendicular to radius PM at M, is the required tangent. Rough figure: a circle with centre P; point M on the circle; radi... 1
- Draw a circle of radius 2.7 cm. Draw a tangent to the circle at any point on it. Answer: Draw the radius to the point and draw a line perpendicular to it at that point; this is the tangent. Steps of construction: Draw a c... 1
- Draw a circle of radius 3.6 cm. Draw a tangent to the circle at any point on it without using the centre. Answer: Use the converse of the tangent-secant angle theorem (alternate segment angle). Analysis: By the converse of the tangent-secant angl... 1
- Draw a circle of radius 3.3 cm. Draw a chord PQ of length 6.6 cm. Draw tangents to the circle at points P and Q. Write your observation about the tangents. Answer: Chord PQ is a diameter (6.6 = 2 × 3.3), so the tangents at P and Q are parallel to each other. Steps of construction: Draw a circle... 1
- Draw a circle with radius 3.4 cm. Draw a chord MN of length 5.7 cm in it. Construct tangents at point M and N to the circle. Answer: Draw radii OM and ON and draw lines perpendicular to them at M and N; these are the tangents (they meet outside the circle). Steps o... 1
- Draw a circle with centre P and radius 3.4 cm. Take point Q at a distance 5.5 cm from the centre. Construct tangents to the circle from point Q. Answer: Draw the circle with diameter PQ; it cuts the given circle at A and B. QA and QB are the tangents (each ≈ 4.3 cm). Steps of construc... 1
- Draw a circle with radius 4.1 cm. Construct tangents to the circle from a point at a distance 7.3 cm from the centre. Answer: Draw a circle on OP (OP = 7.3 cm) as diameter; it cuts the given circle at A and B. PA and PB are the tangents (each ≈ 6 cm). Steps... 1
- The number of tangents that can be drawn to a circle at a point on the circle is ........ (A) 3 (B) 2 (C) 1 (D) 0 Answer: (C) 1 At a point on a circle, the tangent must be perpendicular to the radius at that point. Through a point on a line exactly one p... 1
- The maximum number of tangents that can be drawn to a circle from a point outside it is ........ (A) 2 (B) 1 (C) one and only one (D) 0 Answer: (A) 2 From an external point P, the circle with diameter OP cuts the given circle at exactly two points A and B, so exactly two tang... 1
- If ΔABC ~ ΔPQR and AB/PQ = 7/5, then ........ (A) ΔABC is bigger. (B) ΔPQR is bigger. (C) Both triangles will be equal. (D) Can not be decided. Answer: (A) ΔABC is bigger. For similar triangles all corresponding sides are in the same ratio: AB/PQ=BC/QR=AC/PR=7/5 > 1 . So every side o... 1
- Draw a circle with centre O and radius 3.5 cm. Take point P at a distance 5.7 cm from the centre. Draw tangents to the circle from point P. Answer: Draw the circle with diameter OP; it cuts the given circle at A and B. PA and PB are the tangents (each ≈ 4.5 cm). Steps of construc... 1
- Draw any circle. Take any point A on it and construct tangent at A without using the centre of the circle. Answer: Use the converse of the tangent-secant angle theorem: make the angle between the chord and the line equal to the inscribed angle on... 1
- Draw a circle of diameter 6.4 cm. Take a point R at a distance equal to its diameter from the centre. Draw tangents from point R. Answer: Radius = 3.2 cm, OR = 6.4 cm. The circle on OR as diameter cuts the given circle at A and B; RA and RB are the tangents (each ≈ 5.5... 1
- Draw a circle with centre P. Draw an arc AB of 100° measure. Draw tangents to the circle at point A and point B. Answer: Draw radii PA and PB with ∠APB = 100° and draw perpendiculars to them at A and B; these are the tangents (they meet at 80°). Analysi... 1
- Draw a circle of radius 3.4 cm and centre E. Take a point F on the circle. Take another point A such that E-F-A and FA = 4.1 cm. Draw tangents to the circle from point A. Answer: EA = 3.4 + 4.1 = 7.5 cm. The circle with diameter EA cuts the given circle at P and Q; AP and AQ are the tangents (each ≈ 6.7 cm). S... 1
- ΔABC ~ ΔLBN. In ΔABC, AB = 5.1 cm, ∠B = 40°, BC = 4.8 cm, AC/LN = 4/7. Construct ΔABC and ΔLBN. Answer: In ΔLBN, LB ≈ 8.9 cm (8.925 cm), ∠B = 40°, BN = 8.4 cm. Analysis: AB/LB=BC/BN=AC/LN=4/7 , so ΔLBN is bigger. Vertex B is common, so... 1