maths : Altitudes and Medians of a triangle : Practice Set :4.1

class 8 :maths : Altitudes and Medians of a triangle : Practice Set :4.1Practice Set :4.1 1 .In  ∆LMN, …… is an altitude and …… is a
median. (write the names of appropriate
segments.)ANSWER :In ∆LMN, seg LX is an altitude and seg LY is a median.Question 2.Draw an acute angled ∆PQR. Draw all of its altitudes. Name the point of concurrence as ‘O’.ANSWER :Question 3.Draw an obtuse angled ∆STV. Draw its medians and show the centroid.ANSWER :Question 4.Draw an obtuse angled ∆LMN. Draw its altitudes and denote the ortho centre by ‘O’.ANSWER :Question 5.Draw a right angled ∆XYZ. Draw its medians and show their point of concurrence by G.ANSWER :Question 6.Draw an isosceles triangle. Draw all of its medians and altitudes. Write your observation about their points of concurrence.ANSWER :The point of concurrence of medians i.e. G and that of altitudes i.e. O lie on the same line PS which is the perpendicular bisector of seg QR.Question 7.Fill in the blanks.Point G is the centroid of ∆ABC.i. If l(RG) = 2.5, then l(GC) = ___ii. If l(BG) = 6, then l(BQ) = ____iii. If l(AP) = 6, then l(AG) = ___ and l(GP) = ___.Solution:The centroid of a triangle divides each median in the ratio 2:1.i. Point G is the centroid and seg CR is the median.∴ l(GC)l(RG)=21∴ l(GC)2.5=21 ……[∵ l(RG) = 2.5]∴ l(GC) × 1 = 2 × 2.5∴ l(GC) = 5ii. Point G is the centroid and seg BQ is the median.∴ l(BG)l(GQ)=21∴ 6l(GQ)=21 …..[∵ l(BG) = 6]∴ 6 × 1 = 2 × l(GQ)∴ 62 = l(GQ)∴ 3 = l(GQ)i.e. l(GQ) = 3Now, l (BQ) = l(BG) + l(GQ)∴ l(BQ) = 6 + 3∴ l(BQ) = 9iii. Point G is the centroid and seg AP is the median.∴ l(AG)l(GP)=21∴ l(AG) = 2 l(GP) …..(i)Now, l(AP) = l(AG) + l(GP) … (ii)∴ l(AP) = 2l(GP) + l(GP) … [From (i)]∴ l(AP) = 3l(GP)∴ 6 = 3l(GP) ..[∵ l(AP) = 6]∴ 63 = l(GP)∴ 2 = l(GP)i.e. l(GP) = 2l(AP) = l(AG) + l(GP) …[from (ii)]∴ 6 = l(AG) + 2∴ l(AG) = 6 – 2∴ l(AG) = 4Join us on a journey through the realms of algebra, geometry, calculus, statistics, and beyond. Whether you’re a seasoned mathematician or just starting to dip your toes into the mathematical waters, our engaging and insightful articles cater to all levels of expertise. We’re here to make math not only accessible but also enjoyable for everyone.