NCERT Solutions for Class 7 Maths Chapter 6 - The Triangle and its Properties Exercise 6.1

Exercise 6.1

1. In $\vartriangle PQR$, $D$ is the mid-point of $\overline {QR} $.

$\overline {PM} $ is ___________ 

$PD$ is _______. 

Is $QM = MR$?

Ans: It is given a $\vartriangle PQR$, $D$ is the mid-point of $\ overline {QR} $. It means $QD = DR$. Since $\overline {PM} $ is perpendicular to the side $QR$ of triangle $\vartriangle PQR$, $\overline {PM} $ is the altitude of  $\vartriangle PQR$. Since $QD = DR$, $PD$ is the median of triangle $\vartriangle PQR$.

No, $QM \ne MR$ because $D$ is the mid-point of $\overline {QR} $. It means $QD = DR$.

2. Draw rough sketches for the following:

a) $\vartriangle ABC$, where $BE$ is a median.

Ans: A median of a triangle is a line segment that is drawn from a vertex to the opposite side of the vertex and it divides the opposite side into two equal parts. The rough sketch of $\vartriangle ABC$, where $BE$ is a median, is drawn below.

Here $BE$ is a median in $\vartriangle ABC$ and $AE = EC$.

b) $\vartriangle PQR$, where $PQ$ and $PR$ are the altitudes of the triangle.

Ans: An altitude of a triangle is defined as a perpendicular drawn from the vertex to the line containing the opposite side of the triangle. The rough sketch of $\vartriangle PQR$, where $PQ$ and $PR$ are the altitudes of the triangle is drawn below.

c) $\vartriangle XYZ$, where $YL$ is an altitude in the exterior of the triangle.

Ans: An altitude of a triangle is defined as a perpendicular drawn from the vertex to the line containing the opposite side of the triangle. The rough sketch of $\vartriangle XYZ$, $YL$ is an altitude in the exterior of the triangle is drawn below.

3. Verify by drawing a diagram if the median and altitude of an isosceles triangle can be the same.

Ans: It is given an isosceles triangle. It is required to verify if the median and altitude of the given triangle can be the same. To do this, construct an isosceles triangle. An isosceles triangle has two equal sides. 

Construct an isosceles triangle $\vartriangle ABC$ with sides$AB = AC$ and draw a median $AL$ that divides the base of the triangle into two equal parts.

From the triangle, it can be seen that the median makes a ${90^ \circ }$ angle with the base $BC$. So, $AL$ is the altitude of the triangle $\vartriangle ABC$. Hence verified, $AL$ is the median and altitude of the given triangle $\vartriangle ABC$.

Conclusion

NCERT Class 7 Maths Chapter 6 Exercise 6.1 Solutions clearly explain fundamental concepts of The Triangle and its Properties, such as triangle altitudes and medians. It's important to understand how altitudes determine triangle heights and how medians divide triangles into equal parts. Practicing these ideas improves geometric reasoning and problem-solving skills. Learning these concepts prepares students for more difficult geometry topics and provides a solid foundation for future mathematical studies. Using these answers gives clarity and confidence in using geometric principles successfully.

Class 7 Maths Chapter 6: Exercises Breakdown

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