3 Mark Question

Question 13 Marks

The lengths of the diagonals of a rhombus are 16cm and 12cm respectively. Find the length of each of its sides.

Answer

We know that the diagonals of a rhombus bisect each other at right angles. AC and BD are intersecting at a point O.
$\text{AO}=\frac{1}{2}\text{AC}=\Big(\frac{1}{2}\times16\Big)=8\text{cm}$
$\text{BD}=\frac{1}{2}\text{BD}= \Big(\frac{1}{2}\times12\Big)=6\text{cm}$
From the right $\triangle\text{AOB},$ we have,
$\therefore\text{AB}^2=\text{AO}^2+\text{BO}^2$
$\Rightarrow\text{AB}^2=\big\{(8)^2+(6)^2\big\}\text{cm}^2$
$\Rightarrow\text{AB}=\sqrt{100}=10\text{cm}$
Therefore, length of each side is 10cm. Because all sides of a rhombus are equal.

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Question 23 Marks

In the adjacent figure, ABCD is a rectangle. If BM and DN are perpendiculars from B and D on AC, prove that $\triangle\text{BMC}\cong\triangle\text{DNA}.$ Is it true that BM = DN?

Answer

In $\triangle\text{BMC}$ and $\triangle\text{DNA}:$
$\angle\text{DNA}=\angle\text{BMC}=90^\circ$
$\angle\text{BCM}=\angle\text{DAN}$ (alternative angles)
BC = DA (opposite sides)
By AAS congruency criteria:
$\triangle\text{BMC}\cong\triangle\text{DNA}$ (proved)
So, we can write BM = DN.

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Question 33 Marks

Two adjacent angles of a parallelogram are in the ratio 4 : 5. Find the measure of each of its angles.

Answer

Let the measure of the adjacent angles be 4x and 5x,
$\therefore4\text{x}+5\text{x}=180$
$\Rightarrow9\text{x}=180$
$\Rightarrow\text{x}=20$
Therefore the measue of the required angle is:
$\angle\text{A}=4\times20=80^\circ$
$\angle\text{B}=5\times20=100^\circ$
$\angle\text{B}+\angle\text{C}=180^\circ$
$\Rightarrow\angle\text{C}=180^\circ-100^\circ=80^\circ$
$\angle\text{C}+\angle\text{D}=180^\circ$
$\Rightarrow\angle\text{D}=180^\circ-80^\circ=100^\circ$

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Question 43 Marks

ABCD is a parallelogram in which $\angle\text{A}=110^\circ.$ Find the measure of each of the angles $\angle\text{A},\angle\text{B}$ and $\angle\text{D.}$

Answer

It is a given that ABCD is a parallelogram in which $\angle\text{A}=110^\circ.$ Since, the sum of any two adjacent angle of a parallelogram is 180º, we have,
$\angle\text{A}+\angle\text{B}=180^\circ$
$\Rightarrow\angle\text{B}=180^\circ-110^\circ$
$\Rightarrow\angle\text{B}=706^\circ$
Also, $\angle\text{B}+\angle\text{C}=180^\circ$
$\Rightarrow\angle\text{C}=180^\circ-70^\circ$
$\Rightarrow\angle\text{C}=110^\circ$
Further, $\angle\text{C}+\angle\text{D}=180^\circ$
$\Rightarrow\angle\text{D}=180^\circ-110^\circ$
$\Rightarrow\angle\text{D}=70^\circ$
$\therefore\angle\text{B}=70^\circ,\angle\text{C}=110^\circ$ and $\angle\text{D}=90^\circ.$

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Question 53 Marks

In the adjacent figure, ABCD is a parallelogram and line segments AE and CF bisect the angles A and C respectively. Show that AE || CF.

Answer

In $\triangle\text{AD}$ and $\triangle\text{CBF},$
we have $\text{AD}=\text{BC},\angle\text{B}=\angle\text{D}$ and $\angle\text{DAE}=\angle\text{BCF}$
$\because\angle\text{A}=\angle\text{C}$
$\Rightarrow\frac{1}{2}\angle\text{A}=\frac{1}{2}\angle\text{C}$
$\Rightarrow\angle\text{DAE}=\angle\text{BCF}$
$\therefore\triangle\text{ADE}\cong\triangle\text{CBF}$
And therefore, $\text{CD}-\text{DE}=\text{AB}-\text{BF}$
So, CE = AF
$\therefore$ AECF is a parallelogram,
Hence, AE || CF.

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Question 63 Marks

Two sides of a parallelogram are in the ratio 5 : 3. If its perimeter is 64cm, find the lengths of its sides.

Answer

Let the measure of the sides be 5x and 3x.
Its perimeter = 2(5x + 3x)
$\therefore$ 2(5x + 3x) = 64
⇒ 16x = 64
⇒ x = 4
Therefore, one side = 5 × 4 = 20
Other side = 3 × 4 = 12

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Question 73 Marks

The sum of two opposite angles of a parallelogram is 130°. Find the measure of each of its angles.

Answer

$\angle\text{A}+\angle\text{C}=130$
Let the measure of $\angle\text{A}=\angle\text{C}=\text{x}$
$\therefore2\text{x}=130$
$\Rightarrow\text{x}=65$
Therefore, $\angle\text{A}=65$
$\therefore\angle\text{A}+\angle\text{B}=180$
$\Rightarrow\angle\text{B}=180-65$
$\Rightarrow\angle\text{B}=115$
$\Rightarrow\angle\text{C}=65$
$\therefore\angle\text{C}+\angle\text{D}=180$
$\Rightarrow\angle\text{D}=180-65$
$\Rightarrow\angle\text{D}=115.$

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Question 83 Marks

Two adjacent angles of a parallelogram are (3x - 4)° and (3x + 16)°. Find the value of x and hence find the measure of each of its angles.

Answer

(3x - 4) + (3x + 16) = 180
⇒ 3x - 4 + 3x + 16 = 180
⇒ 6x + 12 = 180
⇒ 6x = 180 - 12
⇒ 6x = 168
⇒ x = 28
Therefore, the measure of the angle is:
$\angle\text{A}=(3\times28-4)=80^\circ$
$\angle\text{B}=(3\times28+16)=100^\circ$

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Question 93 Marks

The perimeter of a parallelogram is 140cm. If one of the sides is longer than the other by 10cm, find the length of each of its sides.

Answer

Let the length of one side be x cm and other is (x + 10)cm.
$\therefore$ 2(x + x + 10) = 140
⇒ 4x + 20 = 140
⇒ 4x = 140 - 20
⇒ 4x = 120
⇒ x = 30
Length of one side is 30cm and other side = (30 + 10) = 40cm.

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