GSEB Class 7 Maths Solutions Chapter 6 ત્રિકોણ અને તેના ગુણધર્મો Exercise 6.1

Question 1. In ΔPQR, D is the midpoint of \(\overline{\mathrm{QR}}\). Identify \(\overline{\mathrm{PM}}\) and \(\overline{\mathrm{PD}}\). Is QM = MR?Answer: In ΔPQR, \(\overline{\mathrm{PM}}\) is the altitude. \(\overline{\mathrm{PD}}\) is the median. No, QM is not equal to MR.
In simple words: For triangle PQR, \(\overline{\mathrm{PM}}\) is the altitude, forming a right angle with base QR. \(\overline{\mathrm{PD}}\) is the median, connecting P to the middle point of QR. Usually, M and D are different places, so the length from Q to M is not typically the same as from M to R.

Exam Tip: Remember that in a general triangle, the altitude and the median from the same vertex to the opposite side are distinct line segments, unless the triangle is isosceles or equilateral where they coincide along the axis of symmetry.

Question 2. Draw rough figures for the following:
(a) In ΔABC, \(\overline{\mathrm{BE}}\) is a median.
(b) In ΔPQR, \(\overline{\mathrm{PQ}}\) and \(\overline{\mathrm{PR}}\) are altitudes of the triangle.
(c) In ΔXYZ, \(\overline{\mathrm{YL}}\) is an altitude lying outside the triangle.Answer: The figures are drawn as instructed for each part.
In simple words: Just draw the shapes as asked. Each drawing shows exactly what the words describe about the triangles.

Exam Tip: When drawing rough figures, focus on accurately representing the specified properties (median, altitude, right angles) rather than exact measurements.

Question 3. Draw a figure and verify whether a median and an altitude can be the same in an isosceles triangle.Answer: In the provided figure, ΔXYZ is an isosceles triangle, meaning sides XY and XZ are equal. Find the middle point P of \(\overline{\mathrm{YZ}}\) and draw the line segment \(\overline{\mathrm{XP}}\), which is its median. Next, by checking with a right-angle tool, the angle \(\angle XPY\) looks like 90°. So, \(\overline{\mathrm{XP}}\) is also an altitude of ΔXYZ. Consequently, \(\overline{\mathrm{XP}}\) acts as both a median and an altitude for the isosceles triangle ΔXYZ.
In simple words: For an isosceles triangle where two sides are equal, the line from the top corner to the middle of the opposite base is both a median and an altitude. It cuts the base exactly in half and forms a perfect right angle.

Exam Tip: The property of the median and altitude coinciding only holds for the vertex angle in an isosceles triangle, or for any vertex in an equilateral triangle.