CBSE Class 10 Triangles Sure Shot Questions Set 03

Question. In a triangle ABC, if AB, BC and AC are the three sides of the triangle, then which of the statements is necessarily true?
(a) AB + BC < AC
(b) AB + BC > AC
(c) AB + BC = AC
(d) \( AB^2 + BC^2 = AC^2 \)
Answer: (b) AB + BC > AC

Question. The sides of a triangle are 12 cm, 8 cm and 6 cm respectively, the triangle is :
(a) acute
(b) obtuse
(c) right
(d) can't be determined
Answer: (b) obtuse

Question. If the angles of a triangle are in the ratio 1 : 4 : 7, then the value of the largest angle is :
(a) 135°
(b) 84°
(c) 105°
(d) None
Answer: (c) 105°

Question. In an equilateral triangle, the incentre, circumcentre, orthocentre and centroid are:
(a) concylic
(b) coincident
(c) collinear
(d) none of these
Answer: (b) coincident

Question. Triangle ABC is such that AB = 9 cm, BC = 6 cm, AC = 7.5 cm. Triangle DEF is similar to \( \triangle ABC \), If EF = 12 cm then DE is :
(a) 6 cm
(b) 16 cm
(c) 18 cm
(d) 15 cm
Answer: (c) 18 cm

Question. In \( \triangle ABC \), AB = 5 cm, AC = 7 cm. If AD is the angle bisector of \( \angle A \). Then BD : CD is:
(a) 25 : 49
(b) 49 : 25
(c) 6 : 1
(d) 5 : 7
Answer: (d) 5 : 7

Question. In a \( \triangle ABC \), AB = AC and \( AD \perp BC \), then :
(a) AB < AD
(b) AB > AD
(c) AB = AD
(d) \( AB \le AD \)
Answer: (b) AB > AD

Question. The difference between altitude and base of a right angled triangle is 17 cm and its hypotenuse is 25 cm. What is the sum of the base and altitude of the triangle is ?
(a) 24 cm
(b) 31 cm
(c) 34 cm
(d) can't be determined
Answer: (b) 31 cm

Question. If AB, BC and AC be the three sides of a triangle ABC, which one of the following is true?
(a) AB – BC = AC
(b) (AB – BC) > AC
(c) (AB – BC) < AC
(d) \( AB^2 – BC^2 = AC^2 \)
Answer: (c) (AB – BC) < AC

Question. If the medians of a triangle are equal, then the triangle is:
(a) right angled
(b) isosceles
(c) equilateral
(d) scalene
Answer: (c) equilateral

Question. The incentre of a triangle is determined by the:
(a) medians
(b) angle bisectors
(c) perpendicular bisectors
(d) altitudes
Answer: (b) angle bisectors

Question. The circumcentre of a triangle is determined by the:
(a) altitudes
(b) median
(c) perpendicular bisectors
(d) angle bisectors
Answer: (c) perpendicular bisectors

Question. The point of intersection of the angle bisectors of a triangle is :
(a) orthocentre
(b) centroid
(c) incentre
(d) circumcentre
Answer: (c) incentre

Question. A triangle PQR is formed by joining the mid-points of the sides of a triangle ABC. 'O' is the circumcentre of \( \triangle ABC \), then for \( \triangle PQR \), the point 'O' is :
(a) incentre
(b) circumcentre
(c) orthocentre
(d) centroid
Answer: (c) orthocentre

Question. If in a \( \triangle ABC \), 'S' is the circumcentre then:
(a) S is equidistant from all the vertices of a triangle
(b) S is equidistant from all the sides of a triangle
(c) AS, BS and CS are the angular bisectors
(d) AS, BS and CS produced are the altitudes on the opposite sides.
Answer: (a) S is equidistant from all the vertices of a triangle

Question. If AD, BE, CF are the altitudes of \( \triangle ABC \) whose orthocentre is H, then C is the orthocentre of :
(a) \( \triangle ABH \)
(b) \( \triangle BDH \)
(c) \( \triangle ABD \)
(d) \( \triangle BEA \)
Answer: (a) \( \triangle ABH \)

Question. In an equilateral \( \triangle ABC \), if a, b and c denote the lengths of perpendiculars from A, B and C respectively on the opposite sides, then:
(a) a > b > c
(b) a > b < c
(c) a = b = c
(d) \( a = c \neq b \)
Answer: (c) a = b = c

Question. What is the ratio of side and height of an equilateral triangle?
(a) 2 : 1
(b) 1 : 1
(c) \( 2 : \sqrt{3} \)
(d) \( \sqrt{3} : 2 \)
Answer: (c) \( 2 : \sqrt{3} \)

Question. The triangle is formed by joining the mid-points of the sides AB, BC and CA of \( \triangle ABC \) and the area of \( \triangle PQR \) is 6 \( cm^2 \), then the area of \( \triangle ABC \) is :
(a) 36 \( cm^2 \)
(b) 12 \( cm^2 \)
(c) 18 \( cm^2 \)
(d) 24 \( cm^2 \)
Answer: (d) 24 \( cm^2 \)

Question. One side other than the hypotenuse of right angle isosceles triangle is 6 cm. The length of the perpendicular on the hypotenuse from the opposite vertex is :
(a) 6 cm
(b) \( 6\sqrt{2} \) cm
(c) 4 cm
(d) \( 3\sqrt{2} \) cm
Answer: (d) \( 3\sqrt{2} \) cm

Question. Any two of the four triangles formed by joining the midpoints of the sides of a given triangle are:
(a) congruent
(b) equal in area but not congruent
(c) unequal in area and not congruent
(d) none of these
Answer: (a) congruent

Question. The internal bisectors of \( \angle B \) and \( \angle C \) of \( \triangle ABC \) meet at O. If \( \angle A = 80^\circ \) then \( \angle BOC \) is :
(a) 50°
(b) 160°
(c) 100°
(d) 130°
Answer: (d) 130°

Question. The point in the plane of a triangle which is at equal perpendicular distance from the sides of the triangle is :
(a) centroid
(b) incentre
(c) circumcentre
(d) orthocentre
Answer: (b) incentre

Question. Incentre of a triangle lies in the interior of :
(a) an isosceles triangle only
(b) a right angled triangle only
(c) any equilateral triangle only
(d) any triangle
Answer: (d) any triangle

Question. In a triangle PQR, PQ = 20 cm and PR = 6 cm, the side QR is :
(a) equal to 14 cm
(b) less than 14 cm
(c) greater than 14 cm
(d) none of these
Answer: (c) greater than 14 cm

Question. The four triangles formed by joining the pairs of mid-points of the sides of a given triangle are congruent if the given triangle is :
(a) an isosceles triangle
(b) an equilateral triangle
(c) a right angled triangle
(d) of any shape
Answer: (d) of any shape

Question. O is orthocentre of a triangle PQR, which is formed by joining the mid-points of the sides of a \( \triangle ABC \), O is :
(a) orthocentre
(b) incentre
(c) circumcentre
(d) centroid
Answer: (c) circumcentre

Question. In a \( \triangle ABC \), a line PQ parallel to BC cuts AB at P and AC at Q. If BQ bisects \( \angle PQC \), then which one of the following relations is always true:
(a) BC = CQ
(b) BC = BQ
(c) \( BC \neq CQ \)
(d) \( BC \neq BQ \)
Answer: (b) BC = BQ

Question. If D is such a point on the side, BC of \( \triangle ABC \) that \( \frac{AB}{AC} = \frac{BD}{CD} \), then AD must be a/an:
(a) altitude of \( \triangle ABC \)
(b) median of \( \triangle ABC \)
(c) angle bisector of \( \triangle ABC \)
(d) perpendicular bisector of \( \triangle ABC \)
Answer: (c) angle bisector of \( \triangle ABC \)

Question. If ABC is a right angled triangle at B and M, N are the mid-points of AB and BC, then \( 4 (AN^2 + CM^2) \) is equal to –
(a) \( 4AC^2 \)
(b) \( 6AC^2 \)
(c) \( 5AC^2 \)
(d) \( \frac{5}{4} AC^2 \)
Answer: (c) \( 5AC^2 \)

Question. If \( \triangle ABC \) and \( \triangle DEF \) are so related that \( \frac{AB}{FD} = \frac{BC}{DE} = \frac{CA}{EF} \), then which of the following is true?
(a) \( \angle A = \angle F \) and \( \angle B = \angle D \)
(b) \( \angle C = \angle F \) and \( \angle A = \angle D \)
(c) \( \angle B = \angle F \) and \( \angle C = \angle D \)
(d) \( \angle A = \angle E \) and \( \angle B = \angle D \)
Answer: (a) \( \angle A = \angle F \) and \( \angle B = \angle D \)

Question. ABC is a right angle triangle at A and AD is perpendicular to the hypotenuse. Then \( \frac{BD}{CD} \) is equal to :
(a) \( \left( \frac{AB}{AC} \right)^2 \)
(b) \( \left( \frac{AB}{AD} \right)^2 \)
(c) \( \frac{AB}{AC} \)
(d) \( \frac{AB}{AD} \)
Answer: (a) \( \left( \frac{AB}{AC} \right)^2 \)

Question. Let ABC be an equilateral triangle. Let \( BE \perp CA \) meeting CA at E, then \( (AB^2 + BC^2 + CA^2) \) is equal to :
(a) \( 2BE^2 \)
(b) \( 3BE^2 \)
(c) \( 4BE^2 \)
(d) \( 6BE^2 \)
Answer: (c) \( 4BE^2 \)

Question. If D, E and F are respectively the mid-points of sides of BC, CA and AB of a \( \triangle ABC \). If EF = 3 cm, FD = 4 cm, and AB = 10 cm, then DE, BC and CA respectively will be equal to :
(a) 6, 8 and 20 cm
(b) 4, 6 and 8 cm
(c) 5, 6 and 8 cm
(d) \( \frac{10}{3}, 9 \) and 12 cm
Answer: (c) 5, 6 and 8 cm

Question. In the right angle triangle \( \angle C = 90^\circ \). AE and BD are two medians of a triangle ABC meeting at F. The ratio of the area of \( \triangle ABF \) and the quadrilateral FDCE is :
(a) 1 : 1
(b) 1 : 2
(c) 2 : 1
(d) 2 : 3
Answer: (a) 1 : 1

Question. ABC is a triangle and DE is drawn parallel to BC cutting the other sides at D and E. If AB = 3.6 cm, AC = 2.4 cm and AD = 2.1 cm, then AE is equal to :
(a) 1.4 cm
(b) 1.8 cm
(c) 1.2 cm
(d) 1.05 m
Answer: (a) 1.4 cm

Question. Consider the following statements:
(1) If three sides of a triangle are equal to three sides of another triangle, then the triangles are congruent.
(2) If three angles of a triangle are equal to three angles of another triangle respectively, then the two triangles are congruent. Of these statements:
(a) 1 is correct and 2 is false
(b) both 1 and 2 are false
(c) both 1 and 2 are correct
(d) 1 is false and 2 is correct
Answer: (a) 1 is correct and 2 is false

Question. ABC is an isosceles triangle with AB = AC. Side BA is produced to D such that AB = AD. Find m \( \angle BCD \).
(a) 60°
(b) 90°
(c) 120°
(d) can't be determined
Answer: (b) 90°

Question. The areas of the similar triangles are in the ratio of 25 : 36. What is the ratio of their respective heights?
(a) 5 : 6
(b) 6 : 5
(c) 1 : 11
(d) 2 : 3
Answer: (a) 5 : 6

Question. A man goes 150 m due east and then 200 m due north. How far is he from the starting point?
(a) 200 m
(b) 350 m
(c) 250 m
(d) 175 m
Answer: (c) 250 m

Question. ABC is a triangle in which \( \angle A = 90^\circ \). \( AN \perp BC \), AC = 12 cm and AB = 5 cm. Find the ratio of the areas of \( \triangle ANC \) and \( \triangle ANB \) :
(a) 125 : 44
(b) 25 : 144
(c) 144 : 25
(d) 12 : 5
Answer: (c) 144 : 25

Question. A vertical stick 15 cm long casts it shadow 10 cm long on the ground. At the same time a flag pole casts a shadow 60 cm long. Find the height of the flag pole.
(a) 40 cm
(b) 45 cm
(c) 90 cm
(d) None
Answer: (c) 90 cm

Question. Vertical angles of two isosceles triangles are equal. Then corresponding altitudes are in the ratio 4 : 9. Find the ratio of their areas :
(a) 16 : 49
(b) 16 : 81
(c) 16 : 65
(d) None
Answer: (b) 16 : 81

Question. In an equilateral triangle of side 2a, calculate the length of its altitude :
(a) \( 2a\sqrt{3} \)
(b) \( a\sqrt{3} \)
(c) \( \frac{a\sqrt{3}}{2} \)
(d) None
Answer: (b) \( a\sqrt{3} \)

Question. In a \( \triangle ABC \), AB = 10 cm, BC = 12 cm and AC = 14 cm. Find the length of median AD. If G is the centroid, find length of GA :
(a) \( \frac{5}{3}\sqrt{7}, \frac{5}{9}\sqrt{7} \)
(b) \( 5\sqrt{7}, 4\sqrt{7} \)
(c) \( \frac{10}{\sqrt{3}}, \frac{8}{3}\sqrt{7} \)
(d) \( 4\sqrt{7}, \frac{8}{3}\sqrt{7} \)
Answer: (d) \( 4\sqrt{7}, \frac{8}{3}\sqrt{7} \)

Question. \( \triangle ABC \) is a right angled triangle at A and AD is the altitude to BC. If AB = 7 cm and AC = 24 cm. Find the ratio of AD is to AM if M is the mid-point of BC.
(a) 25 : 41
(b) 32 : 41
(c) \( \frac{336}{625} \)
(d) \( \frac{625}{336} \)
Answer: (c) \( \frac{336}{625} \)

Question. Area of \( \triangle ABC = 30 \) \( cm^2 \). D and E are the mid-points of BC and AB respectively. Find \( ar(\triangle BDE) \).
(a) 10 cm
(b) 7.5 cm
(c) 15 cm
(d) None
Answer: (b) 7.5 \( cm^2 \)