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JEE • MATHEMATICS • PROPERTIES OF TRIANGLES

Properties of triangles practice · Easy

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Showing 121-150 of 287 questions

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121 Easy
121 of 287
Application +4 | -1
In a ΔABC,  if  b+c=2a\Delta ABC,\;\mathrm{if}\;b + c = 2a and ∠A=60∘\angle A = 60^\circ, then ΔABC\Delta ABC is
View answer and explanation
Correct option: A
Equilateral
Option A (Equilateral) is correct because given the condition b+c=2ab + c = 2a and ∠A=60∘\angle A = 60^\circ, the triangle must be equilateral.
122 Easy
122 of 287
Application +4 | -1
If rr and RR are respectively the radii of the inscribed and circumscribed circles of a regular polygon of nn sides such that Rr=5−1\frac{R}{r}=\sqrt{5}-1, then nn is equal to
View answer and explanation
Correct option: A
5
Option A (5) is correct because by using the formulas for the radii of inscribed and circumscribed circles of a regular polygon, and the given ratio, we can solve for nn and find that n=5n=5.
123 Easy
123 of 287
Application +4 | -1
If in a Δ  ABC,  CD\Delta \;ABC,\;CD is the angular bisector of the ∠  ACB\angle \;ACB, then CDCD is equal to
View answer and explanation
Correct option: C
2aba+bcos⁡C2\frac{{2ab}}{{a + b}}\cos \frac{C}{2}
Option C (2aba+bcos⁡C2\frac{{2ab}}{{a + b}}\cos \frac{C}{2}) is correct because the length of the angular bisector CD is derived using the area of the triangle.
124 Easy
124 of 287
Application +4 | -1
In a triangle ABCABC, if sin⁡A  sin⁡B=abc2\sin A\;\sin B = \frac{{ab}}{{{c^2}}}, then the triangle is
View answer and explanation
Correct option: C
Right angled
Option C (Right angled) is correct because the given condition leads to sin⁡2C=1\sin^2 C = 1, implying that angle C is 90 degrees, hence a right-angled triangle.
125 Easy
125 of 287
Analyzing +4 | -1
In a triangle, if r1=2r2=3r3{r_1} = 2{r_2} = 3{r_3}, then ab+bc+ca\frac{a}{b} + \frac{b}{c} + \frac{c}{a} is equal to
View answer and explanation
Correct option: D
19160\frac{{191}}{{60}}
Option D (19160\frac{{191}}{{60}}) is correct because it's the result of calculating ab+bc+ca\frac{a}{b} + \frac{b}{c} + \frac{c}{a} after finding the ratios of a, b, and c using the given relationship between the exradii.
126 Easy
126 of 287
Application +4 | -1
In ΔABC\Delta ABC, if ∠A=45∘,∠B=75∘\angle A = 45\mathrm{^\circ },\angle B = 75\mathrm{^\circ }, then a+c2a + c\sqrt 2 is equal to
View answer and explanation
Correct option: D
2b2b
Option D (2b2b) is correct because, using the sine rule and given angles, the expression a+c2a + c\sqrt{2} simplifies to 2b2b.
127 Easy
127 of 287
Application +4 | -1
In any △ABC,b2sin⁡2C+c2sin⁡2B\triangle A B C, b^2 \sin 2 C+c^2 \sin 2 B is equal to
View answer and explanation
Correct option: D
4Δ4 \Delta
Option D (4Δ4 \Delta) is correct because the expression simplifies to 4Δ4 \Delta using trigonometric identities and the area formula of a triangle.
128 Easy
128 of 287
Application +4 | -1
In a △ABC,a[bcos⁡C−ccos⁡B]\triangle A B C, a[b \cos C-c \cos B] is equal to
View answer and explanation
Correct option: C
b2−c2b^2-c^2
Option C (b2−c2b^2-c^2) is correct because by applying the cosine rule and simplifying, the expression reduces to b2−c2b^2 - c^2.
129 Easy
129 of 287
Analyzing +4 | -1
If a,ba,b and cc are the sides of a triangle such that a4+b4+c4=2c2(a2+b2){a^4} + {b^4} + {c^4} = 2{c^2}\left( {{a^2} + {b^2}} \right), then the angles opposite to the side CC is
View answer and explanation
Correct option: C
45∘or  135∘45\mathrm{^\circ or}\;135\mathrm{^\circ }
Option C (45∘or  135∘45\mathrm{^\circ or}\;135\mathrm{^\circ }) is correct because by using the cosine rule and the given equation, we can derive that cos⁡2C=12\cos^2 C = \frac{1}{2}, which implies that CC is either 45 or 135 degrees.
130 Easy
130 of 287
Application +4 | -1
In a ABCABC, if b=2,∠B=30∘b = 2,\angle B = 30^\circ, then the area of the circumcircle of ΔABC\Delta ABC in square unit is
View answer and explanation
Correct option: C
4π4\pi
Option C (4π4\pi) is correct because the area of the circumcircle is calculated using the sine rule to find the radius.
131 Easy
131 of 287
Application +4 | -1
In Δ  ABC\Delta \;ABC, if 8R2=a2+b2+c28{R^2} = {a^2} + {b^2} + {c^2}, then the triangle is
View answer and explanation
Correct option: A
Right angled
Option A (Right angled) is correct because the given condition leads to one of the angles being equal to 90 degrees.
132 Easy
132 of 287
Application +4 | -1
In a Δ  ABC,  a=5,  a=\Delta \;ABC,\;a = 5,\;a = and cos⁡(A+B)=3132\cos \left( {A + B} \right) = \frac{{31}}{{32}}. In this triangle, cc is equal to
View answer and explanation
Correct option: D
None of these
Option D (None of these) is correct because the calculated value of c, 3192\frac{{\sqrt {319} }}{2}, does not match any of the provided options.
133 Easy
133 of 287
Application +4 | -1
If OO is the origin and P(2,3)P\left( {2,3} \right) and Q(4,5)Q\left( {4,5} \right) are two points, then OP.OQcos⁡∠POQ=OP.OQ\cos \angle POQ =
View answer and explanation
Correct option: D
23
Option D ((c2−a22ca)2{\left( {\frac{{{c^2} - {a^2}}}{{2ca}}} \right)^2}) is correct because by using the trigonometric identity for sin⁡3B\sin 3B and the given condition 2b2=a2+c22b^2 = a^2 + c^2, the expression sin⁡3Bsin⁡B\frac{{\sin 3B}}{{\sin B}} simplifies to (c2−a22ca)2{\left( {\frac{{{c^2} - {a^2}}}{{2ca}}} \right)^2}.
134 Easy
134 of 287
Application +4 | -1
In a Δ ABC\Delta \,ABC, if b=20, c=21b = 20,\,c = 21 and sin⁡A=35\sin A = \frac{3}{5}, then a=a =
View answer and explanation
Correct option: B
13
Option B (13) is correct because applying the cosine rule with the given values results in a = 13.
135 Easy
135 of 287
Application +4 | -1
In a △ABC\triangle A B C, let ∠C=π2\angle C=\frac{\pi}{2}, if rr is the inradius and RR is the circumradius of the △ABC\triangle A B C, then 2(r+R)2(r+R) equals
View answer and explanation
Correct option: C
a+ba+b
Option C (a+ba+b) is correct because in a right-angled triangle, 2(r+R)=a+b2(r+R) = a+b can be derived using the relationships between inradius, circumradius, and side lengths.

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136 Easy
136 of 287
Application +4 | -1
In a Δ  ABC,cos⁡A+cos⁡B+cos⁡C\Delta \;ABC,\cos A + \cos B + \cos C is equal to
View answer and explanation
Correct option: A
1+rR1 + \frac{r}{R}
Option A (1+rR1 + \frac{r}{R}) is correct because the expression cos⁡A+cos⁡B+cos⁡C\cos A + \cos B + \cos C can be simplified using trigonometric identities and the relationship between inradius (r) and circumradius (R).
137 Easy
137 of 287
Application +4 | -1
The sides of a triangle are respectively 7 cm, 434\sqrt 3 cm and 13\sqrt {13} cm, then the smallest angle of the triangle is
View answer and explanation
Correct option: A
π6\frac{\pi }{6}
Option A (π6\frac{\pi }{6}) is correct because it is the smallest angle calculated using the cosine rule with the smallest side.
138 Easy
138 of 287
Application +4 | -1
In a Δ ABC,  sin⁡  A\Delta \,ABC,\,\,\sin \,\,A and sin⁡  B\sin \,\,B are the roots of the equation c2 x2−c(a+b)x+ab=0{c^2}\,{x^2} - c\left( {a + b} \right)x + ab = 0, then sin⁡  C=\sin \,\,C =
View answer and explanation
Correct option: C
1\mathrm{1}
Option C (1\mathrm{1}) is correct because sin⁡C=1\sin C = 1 is derived directly from the relationship between the sides and angles of the triangle using the given equation.
139 Easy
139 of 287
Application +4 | -1
If b=3,  c=4b = 3,\,\,c = 4 and B=π/3B = \pi /3, then the number of triangles that can be constructed is
View answer and explanation
Correct option: D
Nil
Option D (Nil) is correct because the sine of angle C is greater than 1, which is impossible, indicating that no such triangle can be constructed.
140 Easy
140 of 287
Analyzing +4 | -1
In any △ABC,4(sa−1)(sb−1)(sc−1)\triangle A B C, 4\left(\frac{s}{a}-1\right)\left(\frac{s}{b}-1\right)\left(\frac{s}{c}-1\right) is equal to
View answer and explanation
Correct option: A
rR\frac{r}{R}
Option A (rR\frac{r}{R}) is correct because the given expression simplifies to the ratio of the inradius to the circumradius of the triangle.
141 Easy
141 of 287
Analyzing +4 | -1
We are given b,  cb,\,\,c and sin⁡  B\sin \,\,B such that BB is acute and b<c  sin⁡  Bb < c\,\,\sin \,\,B. Then,
View answer and explanation
Correct option: A
No triangle is possible
Option A (No triangle is possible) is correct because the given condition b<csin⁡Bb < c \sin B implies that sin⁡C>1\sin C > 1, which is impossible.
142 Easy
142 of 287
Application +4 | -1
In a Δ ABC,  cos⁡  C+cos⁡  Ac+a  +  cos⁡  Bb=\Delta \,ABC,\,\,\frac{{\cos \,\,C + \cos \,\,A}}{{c + a}}\,\, + \,\,\frac{{\cos \,\,B}}{b} =
View answer and explanation
Correct option: B
1b\frac{1}{b}
Option B (1b\frac{1}{b}) is correct because by applying sum-to-product identities and sine rule, the expression simplifies to 1b\frac{1}{b}.
143 Easy
143 of 287
Application +4 | -1
A kite is flying with the string inclined at 75∘75^{\circ} to the horizon. If the length of the string is 25 m25 \mathrm{~m}, the height of the kite is
View answer and explanation
Correct option: B
(25/4)(3+1)2(25 / 4)(\sqrt{3}+1) \sqrt{2}
Option B ((25/4)(3+1)2(25 / 4)(\sqrt{3}+1) \sqrt{2}) is correct because it is derived by using the sine of 75 degrees to find the height of the kite.
144 Easy
144 of 287
Application +4 | -1
In triangle ABCA B C if [ΔABC]R=4\frac{[\Delta A B C]}{R}=4, then the value of acos⁡A+bcos⁡B+ccos⁡Ca \cos A+b \cos B+c \cos C, is:
[RR is the circumradius of triangle ABCA B C and [ΔABC][\Delta A B C] is the area of ΔABC]\Delta A B C]
View answer and explanation
Correct option: C
8
Option C is correct because using trigonometric identities and the area formula, the expression simplifies to 8.
145 Easy
145 of 287
Analyzing +4 | -1
If a,  b,  ca,\,\,b,\,\,c denote the sides of a Δ ABC\Delta \,ABC and the equations ax2+bx+c=0a{x^2} + bx + c = 0 and x2+2x+1=0{x^2} + \sqrt 2 x + 1 = 0 have a common root, then ∠ C=\angle \,C =
View answer and explanation
Correct option: B
45∘45^\circ
Option B (45∘45^\circ) is correct because the common root implies that the ratio of coefficients of the two equations are equal, leading to A=C=45∘A = C = 45^\circ.
146 Easy
146 of 287
Application +4 | -1
The sides of a triangle are 4cm, 5cm and 6cm. the area of the triangle is equal to
View answer and explanation
Correct option: B
1547cm2\frac{{15}}{4}\sqrt 7 \mathrm{c}{\mathrm{m}^2}
Option B (1547  cm2\frac{{15}}{4}\sqrt 7 \;\mathrm{c}{\mathrm{m}^2}) is correct because by using Heron's formula with the given side lengths, we can calculate the area of the triangle.
147 Easy
147 of 287
Application +4 | -1
In a triangle ABC,a=5,b=4\mathrm{ABC}, \mathrm{a}=5, \mathrm{b}=4 and cos⁡(A−B)=3132\cos (\mathrm{A}-\mathrm{B})=\frac{31}{32}, then the third side is equal to: (where symbols used have usual meanings)
View answer and explanation
Correct option: C
6
Option C (6) is correct because by using the given information and applying the tangent half-angle formula and cosine rule, we can find the value of the third side.
148 Easy
148 of 287
Application +4 | -1
The sides of a triangle are sin⁡αcos⁡α\sin \alpha \cos \alpha and 1+sin⁡αcos⁡α\sqrt {1 + \sin \alpha \cos \alpha } for some 0<α<π20 < \alpha < \frac{\pi }{2}. Then, the greatest angle of the triangle is
View answer and explanation
Correct option: C
120∘120^\circ
Option C (120°) is correct because, with the given side lengths, the cosine of the greatest angle simplifies to -1/2, implying the angle is 120°.
149 Easy
149 of 287
Application +4 | -1
The angle of elevation of a tower from a point AA due south of it is 30∘30^{\circ} and from a point BB due west of it is 45∘45^{\circ}. If the height of the tower be 100 meters, then AB=A B= ?
View answer and explanation
Correct option: B
200 m200 \mathrm{~m}
Option B (200 m) is correct because it is calculated using trigonometric relationships and the Pythagorean theorem based on the given angles of elevation and the tower's height.
150 Easy
150 of 287
Application +4 | -1
In a triangle the length of the two larger sides are 10 and 9, respectively. If the angles are in A.P., then the length of the third side can be:
View answer and explanation
Correct option: A
91\sqrt {91}
Option A (91\sqrt{91}) is correct because by using the properties of angles in A.P. and the cosine rule, the length of the third side is found to be 91\sqrt{91}.
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