Skip to main content
Back to PROPERTIES OF TRIANGLES

Your Progress

0 / 434 Answered
0
Correct
0
Incorrect
0%
Accuracy
JEE • MATHEMATICS • PROPERTIES OF TRIANGLES

Properties of triangles practice · Mixed difficulty

Choose an answer to check it, or open “View answer” to study the explanation. Sign in to save progress to your account.

Practice format and chapter quiz

These sets use single-correct-answer MCQs. They do not cover every JEE Main or JEE Advanced question format and are not full-length mock papers.

Take a chapter quiz →
Showing 271-300 of 434 questions

Preparing practice and saved progress…

271 Easy
271 of 434
Analyzing +4 | -1
In any Δ ABC\Delta \,ABC, the distance of the orthocentre from the vertices A,B,CA,B,C are in the ratio
View answer and explanation
Correct option: B
cos⁡  A  :cos⁡  B  :cos⁡  C\cos \,\,A\;:\cos \,\,B\;:\cos \,\,C
Option B (cos⁡A:cos⁡B:cos⁡C\cos A:\cos B:\cos C) is correct because the distances of the orthocenter from the vertices of a triangle are proportional to the cosines of the respective angles.
272 Medium
272 of 434
Application +4 | -1
If in a △ABC,∠B=90∘\triangle A B C, \angle B=90^{\circ}, then tan⁡2(A2)\tan ^2\left(\frac{A}{2}\right) is equal to
View answer and explanation
Correct option: A
b−cb+c\frac{b-c}{b+c}
Option A (b−cb+c\frac{b-c}{b+c}) is correct because it is derived by using the definition of cosine and the half-angle formula for tangent in a right-angled triangle.
273 Medium
273 of 434
Analyzing +4 | -1
If A,  B,  C,  DA,\;B,\;C,\;D are the angles of a quadrilateral, then ∑tan⁡A∑cot⁡A\frac{{\sum \tan A}}{{\sum \cot A}} is equal to
View answer and explanation
Correct option: A
∏tan⁡A\prod \tan A
Option A (∏tan⁡A\prod \tan A) is correct because using the condition A+B+C+D=2πA + B + C + D = 2\pi, we can derive the given expression to be equal to the product of tangents.
274 Easy
274 of 434
Analyzing +4 | -1
In a ΔABC,  ∑(b+c)tan⁡A2tan⁡(B−C2)\Delta ABC,\;\sum \left( {b + c} \right)\tan \frac{A}{2}\tan \left( {\frac{{B - C}}{2}} \right) is equal to
View answer and explanation
Correct option: D
0
Option D (0) is correct because the expression simplifies to 0 using trigonometric identities and the sine rule.
275 Easy
275 of 434
Application +4 | -1
In a triangle ABC,if  ∠A=60∘,a=5,b=4ABC,\mathrm{if}\;\angle A = 60^\circ ,a = 5,b = 4, then cc is a root of the equation
View answer and explanation
Correct option: B
c2−4c−9=0{c^2} - 4c - 9 = 0
Option B (c2−4c−9=0{c^2} - 4c - 9 = 0) is correct because it is the quadratic equation derived from applying the cosine rule with the given triangle properties.
276 Easy
276 of 434
Application +4 | -1
If a,b,ca,b,c the sides of a Δ  ABC\Delta \;ABC are in AP and aa is the smallest side, then cos⁡A\cos A equals
View answer and explanation
Correct option: C
4c−3b2c\frac{{4c - 3b}}{{2c}}
Option C 4c−3b2c\frac{{4c - 3b}}{{2c}} is correct because it is derived by using the cosine rule and the given condition that a, b, c are in AP.
277 Medium
277 of 434
Analyzing +4 | -1
In a △ABC,a,b,A\triangle A B C, a, b, A are given and b1,b2b_1, b_2 are two values of the third side bb such that b2=2b1b_2=2 b_1. Then, sin⁡A\sin A is equal to
View answer and explanation
Correct option: B
9a2−c28c2\sqrt{\frac{9 a^2-c^2}{8 c^2}}
Option B (9a2−c28c2\sqrt{\frac{{9 a^2-c^2}}{{8 c^2}}}) is correct because using the properties of quadratic equations and trigonometric identities, we can derive the expression for sin⁡A\sin A.
278 Easy
278 of 434
Application +4 | -1
In a Δ  ABC\Delta \;ABC, if the sides are a=3,b=5a = 3,b = 5 and c=4c = 4, then sin⁡B2+cos⁡B2\sin \frac{B}{2} + \cos \frac{B}{2} is equal to
View answer and explanation
Correct option: A
2\sqrt 2
Option A (2\sqrt{2}) is correct because, given the sides a=3,b=5,c=4a = 3, b = 5, c = 4, angle B is 90 degrees, and sin⁡B2+cos⁡B2=2\sin \frac{B}{2} + \cos \frac{B}{2} = \sqrt{2}.
279 Medium
279 of 434
Application +4 | -1
In a Δ  ABC\Delta \;ABC, angle AA is greater than angle BB. If the measures of angles AA and BB satisfy the equation 3sin⁡x−4sin⁡3x−k=0,    0<k<13\sin x - 4{\sin ^3}x - k = 0,\;\;0 < k < 1, then the measure of angle CC is
View answer and explanation
Correct option: C
2π3\frac{{2\pi }}{3}
Option C (2π3\frac{{2\pi }}{3}) is correct because the given equation implies sin⁡3A=sin⁡3B\sin 3A = \sin 3B, and since A > B, we have A+B=60∘A + B = 60^\circ, leading to C=120∘=2π3C = 120^\circ = \frac{{2\pi }}{3}.
280 Medium
280 of 434
Application +4 | -1
In a triangle the length of the two larger sides are 24\mathrm{24} and 22\mathrm{22}, respectively. If the angles are in AP, then the third side is
View answer and explanation
Correct option: A
12+2 1312 + 2\,\sqrt {13}
Option A (12+2 1312 + 2\,\sqrt {13}) is correct because it is the value of the third side calculated using the properties of triangles with angles in A.P. and the sine rule.
281 Medium
281 of 434
Application +4 | -1
ABCA B C is an equilateral triangle of each side a (>0)(>0). The inradius of the triangle is
View answer and explanation
Correct option: D
a23\frac{a}{2 \sqrt{3}}
Option D (a23\frac{a}{2 \sqrt{3}}) is correct because the inradius of an equilateral triangle is given by the formula a23\frac{a}{2 \sqrt{3}}, where a is the side length.
282 Easy
282 of 434
Understanding +4 | -1
Two triangles are possible if
View answer and explanation
Correct option: B
A<π2,a>csin⁡A,a<cA < \frac{\pi}{2}, a>c \sin A, a < c
Option B (A<π2,a>csin⁡A,a<cA < \frac{\pi}{2}, a>c \sin A, a < c) is correct because it represents the condition where two triangles can be formed given angle A, side a, and side c.
283 Easy
283 of 434
Analyzing +4 | -1
If angles A,  BA,\;B and CC are in AP, then a+bc\frac{{a + b}}{c} is equal to
View answer and explanation
Correct option: B
2cos⁡A−C22\cos \frac{{A - C}}{2}
Option B (2cos⁡A−C22\cos \frac{{A - C}}{2}) is correct because using the sine rule and the given condition that A, B, and C are in AP, the expression simplifies to 2cos⁡A−C22\cos \frac{{A - C}}{2}.
284 Medium
284 of 434
Application +4 | -1
If the angles of a right angled triangle are in A.P., then the ratio of the in-radius and the perimeter is
View answer and explanation
Correct option: C
(2−3)  :2 3\left( {2 - \sqrt 3 } \right)\;:2\,\sqrt 3
Option C is correct because it accurately calculates the ratio of the in-radius to the perimeter of the right-angled triangle with angles in A.P.
285 Easy
285 of 434
Application +4 | -1
In a triangle ABC,(a+b+c)(b+c−a)=kbcA B C,(a+b+c)(b+c-a)=k b c if
View answer and explanation
Correct option: C
0<k<40 < {k} < 4
Option C (0<k<40 < {k} < 4) is correct because the given equation can be transformed using trigonometric identities to show that kk lies between 0 and 4.

Explore Other Chapters in MATHEMATICS

View All →
SEQUENCE OF SERIES 1764 MCQs DEFINITE INTEGRATION 1443 MCQs MATRICES 1363 MCQs INDEFINITE INTEGRATION 1179 MCQs PROBABILITY 1146 MCQs COMPLEX NUMBERS 980 MCQs
286 Hard
286 of 434
Analyzing +4 | -1
There exist a Δ  ABC\Delta \;ABC satisfying
View answer and explanation
Correct option: D
(a+b)2=c2+ab{\left( {a + b} \right)^2} = {c^2} + ab and 2(sin⁡A+cos⁡A)=3\sqrt 2 \left( {\sin A + \cos A} \right) = \sqrt 3
Option D ((a+b)2=c2+ab{\left( {a + b} \right)^2} = {c^2} + ab and 2(sin⁡A+cos⁡A)=3\sqrt 2 \left( {\sin A + \cos A} \right) = \sqrt 3) is correct because it is the only option that satisfies the triangle inequality and angle properties.
287 Medium
287 of 434
Analyzing +4 | -1
In △ABC,a=5,b=4\triangle A B C, a=5, b=4 and cos⁡(A−B)=3132\cos (A-B)=\frac{31}{32}, then side cc is
View answer and explanation
Correct option: A
6
Option A (6) is correct because applying trigonometric identities and the cosine rule leads to the side c being equal to 6.
288 Easy
288 of 434
Application +4 | -1
In any Δ  ABC\Delta \;ABC, if cot⁡  A2  cot⁡  B2,  cot⁡C2\cot \,\,\frac{A}{2}\,\,\cot \,\,\frac{B}{2},\,\,\cot \frac{C}{2} are in A.P., then a,  b,  ca,\,\,b,\,\,c are in
View answer and explanation
Correct option: A
A.P.\mathrm{A}.\mathrm{P}.
Option A is correct because the condition cot⁡A2 cot⁡B2, cot⁡C2\cot \frac{A}{2}\,\cot \frac{B}{2},\,\cot \frac{C}{2} are in A.P. implies that a,b,ca, b, c are in A.P.
289 Easy
289 of 434
Application +4 | -1
In △ABC\triangle A B C, the value of r1+r21+cos⁡C\frac{r_1+r_2}{1+\cos C} is always equal to
View answer and explanation
Correct option: B
2R2 R
Option B (2R2R) is correct because the expression simplifies to 2R2R using trigonometric identities and the properties of exradii.
290 Easy
290 of 434
Application +4 | -1
In Δ  ABC,  (a−b)2cos⁡2C2+(a+b)2sin⁡2C2\Delta \;ABC,\;{\left( {a - b} \right)^2}{\cos ^2}\frac{C}{2} + {\left( {a + b} \right)^2}{\sin ^2}\frac{C}{2} is equal to
View answer and explanation
Correct option: C
c2{c^2}
Option C (c2{c^2}) is correct because the expression simplifies to c2c^2 using trigonometric identities and the cosine rule.
291 Easy
291 of 434
Analyzing +4 | -1
In △ABC,(a+b−c)(b+c−a)(c+a−b)−abc\triangle A B C,(a+b-c)(b+c-a)(c+a-b)-a b c is always
View answer and explanation
Correct option: B
non-positive
Option B (non-positive) is correct because the expression (a+b−c)(b+c−a)(c+a−b)−abc(a+b-c)(b+c-a)(c+a-b)-abc is always less than or equal to zero.
292 Medium
292 of 434
Application +4 | -1
Let a,ba, b and cc be the sides of a triangle and a2+b2+c2ab+bc+ca=P\frac{a^2+b^2+c^2}{a b+b c+c a}=P. Then
View answer and explanation
Correct option: D
1≤P<21 \leq P < 2
Option D (1≤P<21 \leq P < 2) is correct because the ratio a2+b2+c2ab+bc+ca\frac{a^2+b^2+c^2}{ab+bc+ca} is always greater than or equal to 1, and less than 2 for a triangle.
293 Easy
293 of 434
Analyzing +4 | -1
If twice the square of the diameter of a circle is equal to half the sum of the squares of the sides of incribed triangle ABCABC, then sin⁡2A+sin⁡2B+sin⁡2C{\sin ^2}A + \sin^2 B+ {\sin ^2}C is equal to
View answer and explanation
Correct option: C
4
Option C (4) is correct because the given condition leads to the relationship sin⁡2A+sin⁡2B+sin⁡2C=4{\sin ^2}A + {\sin ^2}B+ {\sin ^2}C = 4.
294 Easy
294 of 434
Application +4 | -1
If in a Δ  ABC,  acos⁡2(C2)+ccos⁡2(A2)=3b2\Delta \;ABC,\;a{\cos ^2}\left( {\frac{C}{2}} \right) + c{\cos ^2}\left( {\frac{A}{2}} \right) = \frac{{3b}}{2}, then the sides a,  ba,\;b and cc
View answer and explanation
Correct option: A
Are in AP
Option A (Are in AP) is correct because the given condition simplifies to 2b=a+c2b = a + c, which is the condition for a, b, and c to be in arithmetic progression.
295 Easy
295 of 434
Analyzing +4 | -1
Which of the following pieces of data does not uniquely determine acute angled Δ  ABC\Delta \;ABC(R=R =circumradius)?
View answer and explanation
Correct option: D
a,sin⁡A,  Ra,\sin A,\;R
Option D (a,sin⁡A,  Ra,\sin A,\;R) is correct because knowing aa, sin⁡A\sin A, and RR only gives the ratio bsin⁡B\frac{b}{{\sin B}} or csin⁡(A+B)\frac{c}{{\sin (A + B)}}, and we cannot determine the values of bb, cc, sin⁡B\sin B, and sin⁡C\sin C separately.
296 Easy
296 of 434
Analyzing +4 | -1
The base of a triangle is 80  cm\mathrm{80}\,\,\mathrm{cm} and one of the base angles is 60∘60^\circ. If the sum of the lengths of the other two sides is 90  cm\mathrm{90}\,\,\mathrm{cm}, then the length of the shortest side is
View answer and explanation
Correct option: D
17  cm\mathrm{17}\,\,\mathrm{cm}
Option D (17 cm) is correct because by using the cosine rule and the given conditions, we can find the length of the shortest side to be 17 cm.
297 Easy
297 of 434
Application +4 | -1
If in a Δ  ABC,b+c11=c+a12=a+b13\Delta \;ABC,\frac{{b + c}}{{11}} = \frac{{c + a}}{{12}} = \frac{{a + b}}{{13}}, then cos⁡A\cos A is equal to
View answer and explanation
Correct option: A
15\frac{1}{5}
Option B (-4) is correct because the midpoint of the rectangle's diagonal lies on the given line, allowing us to solve for c.
298 Medium
298 of 434
Analyzing +4 | -1
If in △ABC,∠A=90∘\triangle A B C, \angle A=90^{\circ} and c,sin⁡B,cos⁡Bc, \sin B, \cos B are rational numbers, then
View answer and explanation
Correct option: A
aa and bb are rational
Option A (aa and bb are rational) is correct because given the conditions, it can be shown that both a and b are rational numbers.
299 Easy
299 of 434
Application +4 | -1
In triangle ABC\mathrm{ABC} given 9a2+9b2−17c2=09 a^2+9 b^2-17 c^2=0.If cot⁡A+cot⁡Bcot⁡C=mn\frac{\cot \mathrm{A}+\cot \mathrm{B}}{\cot \mathrm{C}}=\frac{m}{n}, then the value of (m+n)(m+n) equals
View answer and explanation
Correct option: A
13
Option A (13) is correct because by using the Law of Cosines and trigonometric identities, the given equation simplifies to (cot A + cot B) / cot C = 9/4, and thus m+n = 13.
300 Medium
300 of 434
Application +4 | -1
If the ex-radii of a triangle are in H.P., then the corresponding sides are in
View answer and explanation
Correct option: A
A.P.
Option A is correct because if the ex-radii are in H.P., the corresponding sides are in A.P.
Revision notes and key takeaways

Review PROPERTIES OF TRIANGLES in your textbook or class notes, then use this set to check your understanding.

Showing 271-300 of 434 questions

Continue Your Practice

📚 More Sets in This Chapter

Try different difficulty levels to master this topic

→ ENGINEERING MATHEMATICS - PROPERTIES OF TRIANGLES MCQs (Easy) 287 MCQs → ENGINEERING MATHEMATICS - PROPERTIES OF TRIANGLES MCQs (Hard) 7 MCQs → ENGINEERING MATHEMATICS - PROPERTIES OF TRIANGLES MCQs (Medium) 140 MCQs
🎯 Next Chapters to Practice

Continue building your subject mastery

→ SEQUENCE OF SERIES 1764 MCQs → DEFINITE INTEGRATION 1443 MCQs → MATRICES 1363 MCQs → INDEFINITE INTEGRATION 1179 MCQs

Your practice

0 / 434 answered · 434 remaining · 0% accuracy

Progress is stored on this device. Sign in to save to your account.

✓ Correct · ✗ Incorrect · ○ Unanswered

Practice settings

Type at least 2 characters or jump directly:
🩺 NEET MCQs 🔬 JEE MCQs 🎯 JEE College Predictor Study guides 📊 Dashboard
🔓

Save your learning progress

Sign in with a free account to save your practice answers and quiz results, then return to them when you revise.

Continue with Free Account →