Exercise 10.4 Practical-Geometry -NCERT Solutions Class 7

Question 1

Construct \(ΔABC\), given \(m∠A = 60^\circ\), \(m∠B = 30^\circ \) and \(AB = 5.8\,\rm{ cm}\).

Solution

Video Solution

What is known?

Length of side of a \(ΔABC\), \(AB = 5.8\,\rm{ cm}\)  and  \(∠A = 60^\circ\), \(∠B = 30^\circ \).

To construct:

A triangle \(ΔABC\), given \(∠A = 60^\circ\), \(∠B = 30^\circ \) and \(AB = 5.8\,\rm{ cm}\).

Reasoning:

To Construct a \(ΔABC\) such that \(∠A = 60^\circ, ∠B = 30^\circ\) and \(AB = 5.8 \,\rm{cm}\)., follow the steps given below.

Steps:

Steps of construction:

1. Draw a line segment \(AB\) of length \(5.8\,\rm{ cm}.\)
2. At \(A\), draw ray \(AY\) making \(60^\circ\) with \(AB\).
3. At \(B\), draw ray \(BX\) making \(30^\circ\) with \(AB\).
4. Rays \(BX\) and \(AY\) will intersect at point \(C\).

Triangle \(ABC\) is now completed.

Question 2

Construct \(ΔPQR\)  if  \(PQ = 5 \rm{cm}\), \(m∠PQR = 105^\circ \) and \(m∠QRP = 40^\circ \). (Hint: Recall angle-sum property of a triangle).

Solution

Video Solution

What is known?

In triangle \(PQR\) , \(PQ = 5 \rm{cm}\), \(∠PQR = 105^\circ \) and \(∠QRP = 40^\circ \).

To construct:

A triangle \(ΔPQR\) if \(PQ = 5 \rm{cm}\), \(∠PQR = 105^\circ \) and \(∠QRP = 40^\circ \).

Reasoning:

We will use angle-sum property of a triangle to find the measure of \(∠RPQ\)
By angle sum property of a triangle,
\(∠PQR + ∠QRP + ∠RPQ = 180^\circ.\)
\(105^\circ + 40^\circ + ∠RPQ = 180^\circ.\)
So, \(∠RPQ = 35^\circ\)
Now, let’s Construct \(ΔPQR\) such that \(PQ = 5\, {\rm{cm}} , ∠PQR = 105^\circ\) and \(∠RPQ = 35^\circ\), with the steps given below

Steps:

Steps of construction –

1. Draw a line segment \(PQ\) of length \(5 \, \rm{cm}.\)
2. At \(P\), draw \(PX\) making \(35^\circ\) with \(PQ.\)
3. At \( Q,\) draw \(QY\) making an angle of \(105^\circ\) with \(PQ.\)
4. \(PX\) and \(QY\) will intersect at point \(R.\)

\(PQR\) is the required triangle.

Question 3

Examine whether you can construct \(ΔDEF\) such that \(EF = 7.2\,\rm{ cm},\) \(m∠E = 110^\circ\) and \(m∠F = 80^\circ. \)Justify your answer.

Solution

Video Solution

What is known?

Length of one side of a triangle and measure of two angles.

What is unknown?

Whether a triangle be constructed with the given values of length and angles.

Reasoning:

By using the angle sum property, we can find out the third angle. If the angle sum property is followed, then it is possible to construct a triangle and if not then we cannot construct a triangle.

Steps:

We will use angle-sum property of a triangle to find the measure of \(∠D\)

By angle sum property,

\(∠E +∠F + ∠D =\) \(180^\circ .\)

\(110^\circ + 80^\circ + ∠D =\) \(180^\circ. \)

\(190^\circ + ∠D\) \(=\) \(180^\circ .\)

So, \(∠D = -10^\circ\)

Angle of \(-10^\circ\) is not possible, thus we cannot construct triangle \(ΔDEF.\)