GSEB Class 7 Maths Solutions Chapter 6 The Triangles and Its Properties Exercise 6.1

Question 1. In \( \Delta PQR \), D is the mid-point of \( \overline{QR} \).
\( \overline{P M} \) is ______
PD is ______
Is QM \( \neq \) MR?
Answer:

Answer:
\( \overline{P M} \) is an altitude of \( \Delta PQR \).
PD is a median of \( \Delta PQR \).
No, QM \( \neq \) MR.
In simple words: PM represents the height from P to the base QR, so it's an altitude. PD connects P to the midpoint D of QR, making it a median. Since M and D are different points on QR, the segment QM is not the same length as MR.

Exam Tip: Remember that an altitude is a perpendicular line from a vertex to the opposite side, while a median connects a vertex to the midpoint of the opposite side. They are usually different unless the triangle is isosceles or equilateral.

Question 2. Draw rough sketches for the following:
(a) In \( \Delta ABC \), BE is a median.
(b) In \( \Delta PQR \), PQ and PR are altitudes of the triangle.
(c) In \( \Delta XYZ \), YL is an altitude in the exterior of the triangle.
Answer:
(a) In the adjoining figure, \( \overline{B E} \) is a median of \( \Delta ABC \).

(b) In right \( \Delta QPR \), \( \overline{P Q} \) and \( \overline{P R} \) are altitudes of the triangle.

(c) In the following figure, \( \overline{Y L} \) is an altitude of \( \Delta XYZ \).

In simple words: This question asks us to draw different types of lines within a triangle. A median connects a corner to the middle of the opposite side. An altitude is a straight line from a corner that makes a right angle with the opposite side (or its extension). Sometimes, this altitude can be outside the triangle itself, especially with obtuse triangles.

Exam Tip: Pay close attention to definitions for median, altitude, and angle bisector. Practice drawing each type of line in different kinds of triangles (acute, obtuse, right-angled) to understand their positions.

Question 3. Verify by drawing a diagram if the median and altitude of an isosceles triangle can be same.
Answer: In the adjoining figure, \( \Delta ABC \) is an isosceles triangle having AB = AC. Draw its median AD. Now, with the help of a protractor, measure the \( \angle ADB \). We find that \( \angle ADB = 90^\circ \). AD is perpendicular to BC. Thus, \( \overline{A D} \) is the median as well as an altitude of \( \Delta ABC \). Yes, the median and altitude of an isosceles triangle can be the same, particularly from the vertex angle to the base.

In simple words: When a triangle has two sides that are equal (an isosceles triangle), the line that goes from the top corner to the middle of the bottom side is also the line that makes a right angle with the bottom side. So, yes, the median and the altitude can be the same line in an isosceles triangle.

Exam Tip: Remember this special property of isosceles triangles: the median from the vertex angle to the unequal base is also the altitude and angle bisector. This is a common geometric concept to recall.