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Triangle is a three-sided polygon with three edges and three vertices in geometry. The sum of a triangle's interior angles equals 180 degrees is the most significant feature of a triangle and is known as the angle sum property of the triangle.
Types of triangles based on the angle:
- Acute Angled Triangle - They measure less than 90°.
- Obtuse Triangle: They measure greater than 90°.
- Right Triangle: When one of the angles of a triangle is 90°.
Types of triangles based on the sides:
- Scalene triangle: When none of the sides of a triangle are equal.
- Isosceles triangle: When two sides of a triangle are equal.
- Equilateral Triangle: When all three sides of a triangle are equal.
Pythagorean Theorem
The Pythagorean Theorem states that if you draw three squares, one off each of the sides of a right triangle, the area of the two smaller squares equals the area of the larger square. The hypotenuse is the longest side of the triangle. The sum of the lengths of the other two sides, squared, is always equal to the length of that side, squared.
Sample Questions
Ques. D and E are respectively the midpoints on the sides AB and AC of a triangle ABC and BC = 6 cm. If DE || BC, then the length of DE (in cm) is
(A) 2.5
(B) 3
(C) 5
(D) 6
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Ans. B
Explanation. By midpoint theorem,If D and E are respectively the midpoints on the sides AB and AC of a triangle ABC, DE||BC and BC = 6 cm
So, DE will be half of BC i.e. 3cm.
Ques. In triangle PQR, if PQ = 6 cm, PR = 8 cm, QS = 3 cm, and PS is the bisector of angle QPR, what is the length of SR?
(A) 2
(B) 6
(C) 4
(D) 8
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Ans. C
Explanation. Since, PS is the angle bisector of angle QPR. So, by angle bisector theorem,
QS/SR = PQ/PR
⇒ 3/SR = 6/8
⇒ SR = (3 X 8)/6 cm = 4 cm
Ques. The lengths of the diagonals of a rhombus are 16 cm and 12cm. Then, the length of the side of the rhombus is
(A) 9 cm
(B) 10 cm
(C) 8 cm
(D) 20 cm
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Ans: B
Explanation: The diagonals of rhombus bisect each other at right angle, so side of rhombus is the hypotenuse for the triangles formed. Therefore, By Pythagoras theorem
(16/2)2 + (12/2)2 = Side2
⇒ 82 + 62 = Side2
⇒ 64 + 36 = Side2
⇒ Side = 10 cm
Ques. A flag pole 18 m high casts a shadow 9.6 m long. Find the distance of the top of the pole from the far end of the shadow.
(A) 25.6
(B) 20.4
(C) 23.7
(D) 32.5
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Ans: B
Explanation: The far end of shadow is represented by point A, Therefore we need to Find AC. By Pythagoras theorem,
(18)2 + (9.6)2 = (AC)2
⇒ AC2 = 416.16
⇒ AC = 20.4 m (approx)
Ques. Diagonals of a trapezium PQRS intersect each other at the point O, PQ || RS and PQ = 3 RS, Then the ratio of areas of triangles POQ and ROS is:
(A) 1:9
(B) 9:1
(C) 3:1
(D) 1:3
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Ans: B
Explanation: Since,
SR || PQ,
So, ∠OSR= ∠OQP (alternate interior angles)
Also ∠SOR= ∠POQ (vertically opposite angles)
So triangles SOR and POQ are similar,
Therefore,
ar(POQ)/ar(SOR) = (PQ/SR)2
ar(POQ)/ar(SOR) = (3 SR/SR)2
ar(POQ)/ar(SOR) = 9/1
Ques. ABCD is a trapezium in which AB|| DC and P, Q are points on ADand BC respectively such that PQ || DC. If PD = 18 cm, BQ = 35 cm andQC = 15 cm, find AD.
(A) 55cm
(B) 57cm
(C) 60cm
(D) 62cm
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Ans: C
Explanation: Since, ABCD is a trapezium in which AB || DC and P and Q are points on AD and BC, respectively such that PQ || DC. If PD = 18 cm, BQ = 35 cm and QC = 15 cm,
In triangle ABD
DP/AP = OD/OB
In triangle BDC
BQ/QC = OB/OD
This implies
DP/AP = QC/BQ
18/AP = 15/35
AP = (18 x 35)/15
AP = 42
Therefore, AD = AP + DP = 42 + 18 = 60cm
Ques. Areas of two similar triangles are 36 cm2 and 100 cm2. If the length of a side of the larger triangle is 20 cm, then the length of the corresponding side of the smaller triangle is:
(A) 12cm
(B) 13cm
(C) 14cm
(D) 15cm
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Ans: A
Explanation: Let the side of the smaller triangle be x cm.
ar(Larger Triangle)/ar(Smaller Triangle) = (side of larger triangle/side of smaller triangle)2
100/36 = (20/x)2
x = √144
X = 12 cm
Ques. If ABCD is parallelogram, P is a point on side BC and DP when produced meets AB produced at L, then select the correct option
(A) DP/BL = DC/PL
(B) DP/PL = DC/BL
(C) DP/PL = BL/DC
(D) DP/PL = AB/DC
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Ans: B
Explanation: In ΔALD, we have
BP || AD
∴ LB/BA = LP/PD
⇒ BL/AB = PL/DP
⇒ BL/DC = PL/DP [â?µ AB = DC
⇒ DP/PL = DC/BL
Ques. The length of altitude of an equilateral triangle of side 8cm is
(A) √3 cm
(B) 2√3 cm
(C) 3√3 cm
(D) 4√3 cm
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Ans: D
Explanation: The altitude divides the opposite side into two equal parts,
Therefore, BD = DC = 4 cm
In triangle ABD
AB2 = AD2 + BD2
82 = AD2 + 42
AD2 = 64 – 16
AD2 = 48
AD = 4√3 cm
Ques. Corresponding sides of two similar triangles are in the ratio of 2 : 3. If the area of the smaller triangle is 48 cm2, then the area of the larger triangle is:
(A) 108 m2
(B) 107 m2
(C) 106 m2
(D) 230 m2
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Ans: A
Explanation: According to given Question
ar(Larger Triangle)/ar(Smaller Triangle) = (side of larger triangle/side of larger triangle)2
ar(Larger Triangle)/48 = (3/2)2
ar(Larger Triangle) = (9 x 48 )/4
ar(Larger Triangle) = 108 cm2
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