TRIANGLES ONE MORE TIME
1. Two of the medians of a triangle are perpendicular to the opposite sides;
Prove that the triangle is equilateral
2. An equiangular triangle is also equilateral.
3. The straight line which joins the vertex of an isosceles triangle to the midpoint of
the base is perpendicular to the base.
4. The straight line which bisects the vertex angle of an isosceles triangle bisects
also the base.
5. A, B are the two points on the circumference of a circle, centre O; if C be the midpoint of AB, prove that OC is perpendicular to AB.
6. ABCD is a quadrilateral in which AB=AD and CB=CD; prove that angles B and
C are equal.
7. If the opposite sides of a quadrilateral be equal, its opposite angles are also equal.
8. The angles of a rhombus are bisected by its diagonal.
9. Two triangles ABC, BAF on the same side of the same base have their sides AC,
CB respectively equal to the sides BF, FA, and FC is joined; prove that triangles
CAF, FBC are congruent.
10. Two equal straight lines AB and CD are joined towards opposite parts by AD and
BC intersecting in O; if AD=BC, prove that the triangles OAC and OBD are
isosceles.
The Rolling Triangle
Equilateral triangle ABC is sitting on the top of square DEFG so that AC is coincident
with DE. The triangle and square both have edges one unit in length.
Rotate the triangle around point C until point B is coincident with point F and the triangle
is on the right side of the square. Call this one "roll".
Roll the triangle again, this time with B as the center of rotation, putting the triangle on
the bottom of the square. Keep rolling until the triangle is back on top of the square.
What is the length of the path traced out by point A?
Extra: Do the same process, but keep going until the triangle is on top of the square and
the triangle is in its original orientation, with A matched up with D and C matched up
with E. Now how far has point A traveled altogether?
More Extra: If we used a pentagon instead of a square and did the same thing, how
many rolls of the triangle would it take until A returned to its starting point on the
pentagon? How about with a hexagon? Septagon?
Cross Section of a Cube
This problem is a variation on a problem from that GRE preparation book I used last
week, slightly altered because the way they worded this problem was too easy.
A 'cross section' of a cube is a shape that you get when you cut the cube with a plane sort of like slicing it.
You have a cube with a surface area of 96 cm^2 (that's 96 square centimeters). If you cut
the cube with a plane that is parallel to one of its faces, you will get a square.
Questions:
1. What would be the perimeter of that square?
2. What is the perimeter of the largest rectangle you can get as a cross
section?
3. Can you figure out how to get an equilateral triangle as a cross section?
Extra: What's the area of the square? the rectangle? the biggest possible equilateral
triangle?
To make explaining this easier, I've provided a labelled cube below. ABCD is the front of
the cube, while EFGH is the back. Be sure to explain your answers so that someone else
might learn something from reading your solution.
3. In how many different ways can you divide a
square into 4 congruent shapes?
4. A bug crawls around the outside of a rectangle
that measures 6.5 cm by 8.5 cm. If the bug stays
3.5 cm from the rectangle at all times, what is the
area, to the nearest cm squared, of the region
bound by the path of the bug?