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Isosceles Triangle Worksheet

Isosceles Triangle Worksheet
  • Page 1
 1.  
If the perimeter of an isosceles triangle is 12 times its base, then what is the ratio of the length of its leg to base?
a.
4 : 1
b.
2 : 11
c.
1 : 4
d.
11 : 2


Solution:

Let x be the length of the base and y be the length of two equal sides.

x + 2 y = 12x

11x = 2y

yx = 11 / 2

Ratio of the length of leg to base = 11 : 2


Correct answer : (4)
 2.  
mABC = 57. Find the measure of ACB.


a.
20
b.
19
c.
28
d.
58


Solution:

mACB = mDBC
[BD = DC.]

mADB = 2mDBC = 2mACB
[Exterior angles.]

mADB = 1 / 2 × (180 - mBAD)
[AB = AD.]

mBAD = 180 - (mABC + mACB)
[Angles in the same triangle.]

mBAD = 114 - mACB
[mABC = 57.]

mADB = 1 / 2 × (180 - (114 - mACB)
[Step 3.]

mADB = 1 / 2 × (57 + mACB)
[Step 3.]

2mDCB = 1 / 2 × (57 + mACB)
[Steps 3 and 7.]

2mACB = 1 / 2 × (57 + mACB)
[DCB = ACB.]

mACB = 19
[Solve.]


Correct answer : (2)
 3.  
ΔABC is an isosceles triangle and x = 40. Select the correct statement/statements.
1. Base is longer than the legs.
2. m ACD = 110



a.
2 only
b.
1 only
c.
both 1 and 2
d.
both are wrong


Solution:


[In ΔABC, AB and AC are legs.]

mBAC + mABC + mACB = 180
[Sum of the measures of the angles in a triangle.]

mABC = 1 / 2(180 - 40) = 70
[mABC = mACB.]

mBAC < mABC
[Step 3.]

BC < AC
[Side opposite to larger angle will be larger.]

Base is shorter than legs.
[Step 5.]

mACD = 180 - mACB
[Supplementary angles.]

= 180 - 70 = 110

Statement 2 only is correct.


Correct answer : (1)
 4.  
The angles 1 and 2 are in the ratio 3 : 4. Find m3.

a.
54.435
b.
48.435
c.
51.435
d.
56.435


Solution:

m1 : m2 = 3 : 4
[Given.]

m1 + m2 = 180
[Supplementary angles.]

Let m1 = 3x, then m2 = 4x
[Angles are in the ratio 3 : 4.]

3x + 4x = 180
[Step 2.]

x = 25.71
[Solve.]

m1 = 3x = 77.13

m3 = 1 / 2(180 - m1)
[m3 = m4.]

= 51.435


Correct answer : (3)
 5.  
Which of the following is/are true?
1. If AO = OD, then ABDC will be a rhombus
2. If AD = BC, then ABDC will be a square
3. BC is always the perpendicular bisector of AD
4. AD is always the perpendicular bisector of BC


a.
All are correct
b.
1, 2 & 4 only
c.
1, 2 & 3 only
d.
1 & 2 only


Solution:

In ΔABC, AB = AC, AO ^ BC, OB = OC
[Isosceles triangle theorem.]

AB = AC B = C.

When AO = OD, AB = AC = BD = CD; ABDC becomes a rhombus.
[All the sides are equal.]

When AD = BC, AO = OB, mBAO = 45, mBAC = 90, ABDC becomes a square.

BC need not be the perpendicular bisector of AD always. BC will be perpendicular to AD always. BC will be the perpendicular bisector only if AB = BD.

AD will be the perpendicular bisector of BC always as AB = AC.

Only statements 1, 2 and 4 are correct.


Correct answer : (2)
 6.  
What shall be the value of x so that AE will be parallel to BC, if y = 106?


a.
72
b.
76
c.
74
d.
78


Solution:

mDBC = 106
[Given.]

mDBC = mBAE
[Corresponding Angles.]

mABC = mACB = 74
[DBC & ABC are Supplementary angles, AB = AC, mABC = mACB.]

mBAC = 180 - 2 × 74 = 32

x = 106 - 32 = 74
[Step 2.]


Correct answer : (3)
 7.  
In ΔABC, BD : DC is

a.
1 : 1
b.
2 : 1
c.
1 : 2
d.
3 : 1


Solution:

In an Isosceles triangle, the perpendicular drawn from the vertex onto the base, bisects the base.

AD BC BD = DC
[Step 1.]

BD : DC = 1 : 1


Correct answer : (1)
 8.  
In the figure, if BAD = 31o, then DAC equals

a.
33o
b.
31o
c.
29o
d.
59o


Solution:

In an Isosceles triangle, the perpendicular drawn from the vertex to the base is the bisector of the vertex angle.

AD BC BAD = DAC
[Step 1.]

DAC = 31o
[BAD = 31o.]


Correct answer : (2)
 9.  
In ΔABC, if x = 64, then find mB.

a.
58
b.
68
c.
63
d.
53


Solution:

AB = AC B = C
[Isosceles triangle theorem.]

mA + mB + mC = 180
[Triangle-Angle-Sum theorem.]

64 + mB + mB = 180
[C = B.]

2mB = 116
[Simplify.]

mB = 58
[Solve for B.]


Correct answer : (1)
 10.  
In ΔABC, mA = mB. Find the perimeter of ΔABC if x = 8 cm and y = 10 cm.


a.
26 cm
b.
32 cm
c.
28 cm
d.
34 cm


Solution:


In ΔABC, mA = mB AC = BC
[Converse of Isosceles triangle theorem.]

Perimeter of ΔABC = AB + BC + CA
[Definition.]

Perimeter of ΔABC = 8 + 10 + 10 = 28 cm
[Substitute and add.]


Correct answer : (3)

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