Triangles and Polygons | Problems 2
*Problems Part 2
- Two triangles are congruent if two sides and the enclosed median in one triangle are respectively equal to two sides and the enclosed median of the other.
- The two sides AB, AC in the triangle ABC are produced to D, E respectively so that BD = BC = CE. If BE and CD intersect in F, show that ∠BFD = right angle – ½∠A.
- From any point D on the base BC of an isosceles triangle ABC, a perpendicular is drawn to cut BA and CA or produced in M and N. Prove that AMN is an isosceles triangle.
- ABC is a triangle in which AB is greater than AC. If D is the middle point of BC, then the angle CAD will be greater than the angle BAD.
- The straight line drawn from the middle point of the base of a triangle at right angles to the base will meet the greater of the two sides, not the less.
- ABC is a right-angled triangle at A. The altitude AD is drawn to the hypotenuse BC. DA and CB are produced to P and Q respectively so that AP = AB and BQ = AC. Show that CP = AQ.
- A square is described on the hypotenuse BC of a right-angled triangle ABC on the opposite side to the triangle. If M is the intersection of the diagonals of the square and LMN is drawn perpendicular to MA to meet AB, AC produced in L, N respectively, then BL = AC,CN = AB.
- Show that the sum of the altitudes of a triangle is less than the sum of its three sides.
- On BC as a base, an equilateral triangle ABC and an isosceles triangle BBC are drawn on the same side of BC such that ∠D = half ∠A.Prove that AB = BC.
- P is any point inside or outside the triangle ABC. AP, BP, CP are produced to R, S, T respectively so that AP = PR, BP = PS, CP = PT. Show that the triangles RST and ABC are equiangular.
- The interior and exterior angles at C of a triangle ABC are bisected by CD,CF to meet AB and BA produced in D, F respectively. From D a line BR is drawn parallel to BC to meet AC in R. Show that FR produced bisects BC. (Produce DR to meet CF in S.)
- ABC is an isosceles triangle in which AB = AC. On AB a point G is taken and on AC produced the distance CH is taken so that BG = CH. Prove that GH > BC.
- Construct a triangle having given the base, the difference of the base angles, and the difference of the other two sides. [P1-10]
- Show that the sum of the three medians in a triangle is less than its perimeter and greater than ¾ the perimeter.
- ABC is an isosceles triangle and D any point in the base BC; show that perpendiculars to BC through the middle points of BD and DC meet AB, AC in points H, K respectively so that BH = AK and AH = CK. (Join BH, BK.)
- The side AB of an equilateral triangle ABC is produced to D so that BD = 2 AB. A perpendicular DF is drawn from D to CB produced. Show that FAC is a right angle.
- On the two arms of a right angle with vertex at A, AB is taken = AD and also AC = AE, so that B, C are on the same area of ∠A. Prove that the perpendicular from A to CD when produced bisects BE.