Plane Geometry problems
- 1. In ?ABC, the angle bisectors of ?B and ?C meet at O. If ?A = 70°, then ?BOC is equal to:
Options- A. 135°
- B. 125°
- C. 115°
- D. 110° Discuss
Correct Answer: 125°
Explanation:
- 2. The complement of an angle exceeds the angle by 60°. Then the angle is equal to:
Options- A. 25°
- B. 30°
- C. 15°
- D. 35° Discuss
Correct Answer: 15°
Explanation:
Let, the angle be A
? Its complement angle = 90° ? A
According to the question
(90 ? A) = A + 60°
? 90 - 60 = A + A
? 2A = 30°
? A = 15°
- 3. In the given figure, AB || CD and AC || BD. If ?EAC = 40°, ?FDG = 55°, ?HAB = x; then the value of x is:
Options- A. 95°
- B. 70°
- C. 35°
- D. 85° Discuss
Correct Answer: 85°
Explanation:
?DCK = ?FDG = 55° (corr. ?s)
? ?ACE = ?DCK = 55° (vert. opp. ?s)
So, ?AEC = 180° ? (40° + 55°) = 85°
? ?HAB = ?AEC = 85° (corr. ?s)
Hence, x = 85°.
- 4. In the given figure, OP bisect ?BOC and OQ bisects ?AOC. Then ?POQ is equal to :
Options- A. 90°
- B. 120°
- C. 60°
- D. 100° Discuss
Correct Answer: 90°
Explanation:
Since OP bisects ?BOC,
? ?BOC = 2?POC
Again, OQ bisects ?AOC,
? ?AOC = 2?QOC
Since ray OC stands on line AB, ?,
?AOC + ?BOC = 180°
? 2?QOC + 2?POC = 180°
? 2?QOC + ?POC = 180°
? ?QOC + ?POC = 90°
? ?POQ = 90°.
The above sum can also be restated as follows; The angle between the bisectors of a linear pair of angles is a right angle.
- 5. If two parallel lines are intersected by a transversal, then the bisectors of the two pairs of interior angles enclose a:
Options- A. Trapezium
- B. Rectangle
- C. Square
- D. circle Discuss
Correct Answer: Rectangle
Explanation:
- 6. In the given figure, ?B = ?C = 55° and ?D = 25°. Then:
Options- A. BC < CA < CD
- B. BC > CA > CD
- C. BC < CA, CA > CD
- D. BC > CA, CA < CD Discuss
Correct Answer: BC > CA, CA < CD
Explanation:
According to question,
?B = ?C = 55° , ?D = 25°
We can say ,
AB = AC ( ? ?B = ?C = 55° )
In Triangle ABC,
?A + ?B + ?C = 180°
? ?A + 55° + 55° = 180°
? ?A + 110° = 180°
? ?A = 180° - 110°
? ?A = 70° ..........................(1)
As per given figure,
?ACD + ?ACB = 180° ( ?ACB = ?C = 55°)
? ?ACD + 55° = 180°
? ?ACD = 180° - 55°
? ?ACD = 125° ....................... (2)
Now in Triangle ACD,
?CAD + ?ACD + ?CDA = 180°
? ?CAD + 125° + 25° = 180°
? ?CAD + 150° = 180°
? ?CAD = 30° ...........................(3)
( In a ?, greater angle has longer side opposite to it )
From the equation (1) , (2) and (3);
?B < ?A and ?CAD > ?D ;
? BC > CA and CA < CD
- 7. In the given figure, AM ? BC and AN is the bisector of ?A. What is the measure of ?MAN.
Options- A. 17.5°
- B. 15.5°
- C. 20°
- D. 25° Discuss
Correct Answer: 17.5°
Explanation:
- 8. In the given figure ?QPR = 90°, QR = 26 cm, PM = 6 cm, MR = 8 cm and ?PMR = 90°, find the area of ?PQR.
Options- A. 180 cm2
- B. 240 cm2
- C. 120 cm2
- D. 150 cm2 Discuss
Correct Answer: 120 cm2
Explanation:
Given in the question, QR = 26 cm, PM = 6 cm, MR = 8 cm
According to question, in ? PMR
(Pythagoras Theorem)
PR = ?PM2 + MR2 = ?36 + 64 = ? 100 = 10 cm
According to question, in ? PQR
(Pythagoras Theorem)
PQ = ?QR2 - PR2 = ?262 - 102 = ?576 - 100 = ? 476 = 24
? area of triangle ?PQR = Base length x Height / 2
? area of triangle ?PQR = PR x PQ / 2
? area of triangle ?PQR = 10 x 24 / 2 = 10 x 12
? area of triangle ?PQR = 120
- 9. In the given figure, find the length of BD.
Options- A. 13.5 cm
- B. 12 cm
- C. 14.5 cm
- D. 15 cm Discuss
Correct Answer: 13.5 cm
Explanation:
- 10. In, ?ABC, D and E are the mid-points of AB and AC respectively. Find the ratio of the areas of ?ADE and ?ABC.
Options- A. 1 2
- B. 1 4
- C. 3 4
- D. 1 8 Discuss
Correct Answer: 1 4
Explanation:
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