All questions

• Ratio of areas of two triangles with equal heights is 2 : 3. If base of the smaller triangle is 6 cm then what is the corresponding base of the bigger triangle? Answer: The base of the bigger triangle is 9 cm. Areas of triangles with equal heights are proportional to their bases. Let the base of the... Similarity 1 answer
• In figure 1.73, ∠ABC = ∠DCB = 90°, AB = 6, DC = 8 then A(ΔABC)/A(ΔDCB) = ? Answer: A(triangle ABC)/A(triangle DCB) = 3/4 Both triangles have the same base BC. Their heights on this base are AB = 6 (since AB ⊥ BC) an... Similarity 1 answer
• In figure 1.74, PM = 10 cm, A(ΔPQS) = 100 sq.cm, A(ΔQRS) = 110 sq.cm then find NR. Answer: NR = 11 cm triangle PQS and triangle QRS have the common base QS. Their heights are PM and NR respectively. Areas of triangles with... Similarity 1 answer
• ΔMNT ~ ΔQRS. Length of altitude drawn from point T is 5 and length of altitude drawn from point S is 9. Find the ratio A(ΔMNT)/A(ΔQRS). Answer: A(triangle MNT)/A(triangle QRS) = 25/81 Since triangle MNT sim triangle QRS , vertex T corresponds to vertex S, so the altitudes fro... Similarity 1 answer
• In figure 1.75, A-D-C and B-E-C, seg DE ∥ side AB. If AD = 5, DC = 3, BC = 6.4 then find BE. Answer: BE = 4 Let BE = x, then EC = 6.4 − x. In triangle ABC , DE ∥ AB, so by the basic proportionality theorem, CD/DA = CE/EB 3/5 = 6.4 -... Similarity 1 answer
• In the figure 1.76, seg PA, seg QB, seg RC and seg SD are perpendicular to line AD. AB = 60, BC = 70, CD = 80, PS = 280 then find PQ, QR and RS. Answer: PQ = 80, QR = 280/3 , RS = 320/3 Lines perpendicular to the same line are parallel, so PA ∥ QB ∥ RC ∥ SD. Parallel lines cut two tra... Similarity 1 answer
• In ΔPQR seg PM is a median. Angle bisectors of ∠PMQ and ∠PMR intersect side PQ and side PR in points X and Y respectively. Prove that XY ∥ QR. Answer: By the angle bisector theorem PX/XQ = MP/MQ and PY/YR = MP/MR ; since MQ = MR these are equal, so XY ∥ QR. In triangle PMQ , ray MX... Similarity 1 answer
• In fig 1.78, bisectors of ∠B and ∠C of ΔABC intersect each other in point X. Line AX intersects side BC in point Y. AB = 5, AC = 4, BC = 6 then find AX/XY. Answer: AX/XY = 3/2 In triangle ABY , BX bisects ∠ B , so by the angle bisector theorem AX/XY = AB/BY In triangle ACY , CX bisects ∠ C , so... Similarity 1 answer
• In □ABCD, seg AD ∥ seg BC. Diagonal AC and diagonal BD intersect each other in point P. Then show that AP/PD = PC/BP. Answer: triangle APD sim triangle CPB (AA test), which gives AP/PD = PC/BP . In triangle APD and triangle CPB : ∠ PAD cong ∠ PCB (alternate... Similarity 1 answer
• In fig 1.80, XY ∥ seg AC. If 2AX = 3BX and XY = 9. Complete the activity to find the value of AC. Answer: AC = 22.5 2AX = 3BX, so AX/BX = 3/2 . By componendo, AX + BX/BX = 3 + 2/2 , so AB/BX = 5/2 …… (I) In triangle BCA and triangle BYX ,... Similarity 1 answer
• In figure 1.81, the vertices of square DEFG are on the sides of ΔABC. ∠A = 90°. Then prove that DE² = BD × EC. (Hint: Show that ΔGBD is similar to ΔCFE. Use GD = FE = DE.) Answer: triangle GBD sim triangle CFE (AA test), so GD/CE = BD/FE ; with GD = FE = DE this gives DE² = BD × EC . DEFG is a square, so ∠ GDB... Similarity 1 answer
• In figure 2.17, ∠MNP = 90°, seg NQ ⊥ seg MP, MQ = 9, QP = 4, find NQ. Answer: NQ = 6 In right-angled triangle MNP , NQ is the perpendicular from the right angle to the hypotenuse. By the theorem of geometric me... Pythagoras Theorem 1 answer
• See figure 2.19. Find RP and PS using the information given in ΔPSR. (In ΔPSR, ∠S = 90°, ∠P = 30°, RS = 6) Answer: RP = 12 and PS = 6√(3) triangle PSR is a 30^° – 60^° – 90^° triangle ( ∠ R = 60^° ), with hypotenuse RP. Side opposite the 30^° angl... Pythagoras Theorem 1 answer
• For finding AB and BC with the help of information given in figure 2.20, complete following activity. (In ΔABC, ∠B = 90°, AB = BC, AC = √8) Answer: AB = BC = 2 AB = BC …… given So triangle ABC is an isosceles right-angled triangle and ∠ BAC = ∠ BCA = 45^° . In a 45^° – 45^° – 90^... Pythagoras Theorem 1 answer
• Find the side and perimeter of a square whose diagonal is 10 cm. Answer: Side = 5√(2) cm, perimeter = 20√(2) cm Let the side be x cm. The diagonal divides the square into two right-angled isosceles triangl... Pythagoras Theorem 1 answer
• In figure 2.21, ∠DFE = 90°, FG ⊥ ED, If GD = 8, FG = 12, find (1) EG (2) FD and (3) EF Answer: (1) EG = 18 (2) FD = 4√(13) (3) EF = 6√(13) (1) By the theorem of geometric mean, FG² = EG × GD 144 = EG × 8 ⇒ EG = 18 (2) In right-... Pythagoras Theorem 1 answer
• Find the diagonal of a rectangle whose length is 35 cm and breadth is 12 cm. Answer: The diagonal is 37 cm. The diagonal is the hypotenuse of a right-angled triangle whose sides are the length and breadth. By Pythagor... Pythagoras Theorem 1 answer
• In the figure 2.22, M is the midpoint of QR. ∠PRQ = 90°. Prove that, PQ² = 4PM² − 3PR² Answer: Using Pythagoras theorem in triangle PRQ and triangle PRM with QR = 2RM gives PQ² = 4PM² - 3PR² . M is the midpoint of QR, so QR = 2... Pythagoras Theorem 1 answer
• In ΔABC, AB = 10, AC = 7, BC = 9 then find the length of the median drawn from point C to side AB Answer: The median is 2√(10) . Let CM be the median, so M is the midpoint of AB and AM = 5. By Apollonius theorem, CA² + CB² = 2CM² + 2AM² 4... Pythagoras Theorem 1 answer
• In the figure 2.28 seg PS is the median of ΔPQR and PT ⊥ QR. Prove that, (1) PR² = PS² + QR × ST + (QR/2)² (2) PQ² = PS² − QR × ST + (QR/2)² Answer: Apply Pythagoras theorem in triangle PTR and triangle PTQ , writing TR = TS + SR and QT = QS − TS, with QS = SR = QR/2. S is the mid... Pythagoras Theorem 1 answer
• In ΔABC, point M is the midpoint of side BC. If, AB² + AC² = 290 cm², AM = 8 cm, find BC. Answer: BC = 18 cm AM is a median. By Apollonius theorem, AB² + AC² = 2AM² + 2BM² 290 = 2 × 64 + 2BM² 2BM² = 162, BM² = 81 BM = 9 BC = 2BM =... Pythagoras Theorem 1 answer
• In figure 2.30, point T is in the interior of rectangle PQRS, Prove that, TS² + TQ² = TP² + TR² (As shown in the figure, draw seg AB ∥ side SR and A-T-B) Answer: Since AB ⊥ PS and AB ⊥ QR, apply Pythagoras theorem in the four right triangles at A and B, and use AP = BQ, AS = BR. AB ∥ SR ∥ PQ,... Pythagoras Theorem 1 answer
• Out of the following which is the Pythagorean triplet? (A) (1, 5, 10) (B) (3, 4, 5) (C) (2, 2, 2) (D) (5, 5, 2) Answer: (B) (3, 4, 5) 3² + 4² = 9 + 16 = 25 = 5² , so the square of the largest number equals the sum of the squares of the other two. Check... Pythagoras Theorem 1 answer
• In a right angled triangle, if sum of the squares of the sides making right angle is 169 then what is the length of the hypotenuse? (A) 15 (B) 13 (C) 5 (D) 12 Answer: (B) 13 By Pythagoras theorem, (hypotenuse) ² = sum of the squares of the sides making the right angle = 169. hypotenuse = √(169) = 1... Pythagoras Theorem 1 answer