Question

21. Sketch a scalene triangle and draw the three midsegments. The three midsegments of a triangle divide it into four smaller triangles. a. Why do the four smaller triangles have equal area? b. What is the ratio of the perimeter of one of the smaller triangles to the perimeter of the original triangle? Justify your answer.

Answer 1

Hence, in the four triangles D APR, DPBQ, DPUR and OROC are are equal to h Base of all the triangles equal to y Altitude of all triangles the area of all the four triangles are equal (proved.) (6) Perimeter of the original DABLE ABA BC + AC = 22+ 2y +22 salo+y+z) Perimeter of the DAPRE AP+ PRTAR = 1 AB + 2 BC + LAC s xd ytz Similarly perimeter of the other smaller triangles are Detytz O ty+2 Persimeter of one of the smaller triangle Perimeter of the original triangle 2(x+9+2) 5 놀 the ratio of the perimeter of the one of the smaller triangle to the perimeter of the original triangle is 1/2 07 1:2 (answa)

Solution: Concepts In DABC, А. If p and a are the mid-points of sides AB and AC respectively then D ☺ POS 11 BC (ii) Po intersects AD Caltitude) at mid-point ine. Pe bisects AD :. AR =RD (iii) PQ = £ BC (iv) AR is perpendicular to Pu il. AR is an altitude to the base pu from A in DAPY. (: POSIBC,, LARUS = CROC=909)

an 1 coe Let A ABC is scalene triangle. And P, Q and R the mid-points of the sides AB, BC and CA respectively as shown in the figure below. Here, AB CACCBC R Let AB = 2x BC =24 B C AC = 22 As=2h .. BCS Ecus y (c is mid-poind) from concept (iii)) PR = ££ BC = (29) ... PREY PR bisects AS ( boomi concepte (vi)) . ATE TS AT = LAS = Ź (24) سا . ATEL AT is an altitude (boom concept (1) OPBCS, OP CR and ARU =AS = h Altitude of TS =