Study Questions on Similar Triangles

Some Problems about Similarity

These are all problems that can be solved by similar figures. Some can be solved other ways as well (e.g. Pythagorean theorem, intersecting chords, ...)

Distances on rays OA and OB

Given points O, A, B, suppose that |OA| = a and |OB| = b, and that A' is a point on ray OA with |OA'| = 1/a and B' is a point on ray OB with |OB'| = 1/b.

If |AB| = c, what is |A'B'|?

Your answer should be in terms of a, b, c and you should give a brief but convincing explanation. (If you are not sure of your answer, construct by and and measure a simple example or two to get an experimental check of your work.)

Some special points in a Right Triangle

Here are a couple of problems about a right triangle FGH with right angle at F and sides f = GH, g = HF, h = FG. In addition to computing the answer for the general case below, try a numerical example, such as the 3-4-5 right triangle and check your answers experimentally.

Note: This problem is closely linked to the next one about isosceles triangles.

Compute h in terms of lengths f and g.

If the perpendicular bisector of GH intersects GH at M and FG at J, compute GJ in terms of lengths f and g.

If the angle bisector of angle GHF intersects FG at K, compute FK in terms of lengths f and g. (This one requires knowing something about the ratio in which an angle bisector cuts an opposite side.)

Centers and Radii of Isosceles

Given an isosceles triangle ABC, with |AB| = |AC| = b and |BC| = a.

What is the radius of the circumcircle of ABC? (The answer should be in terms of a and b.)
What is the radius of the inscribed circle of ABC?

A distance relationship from a circle

In the figure, the circle has center O and radius r.

PQ and NS are diameters.
Line P1 Q1 is tangent to the circle at S.
(N, P and P1 are collinear and also N, Q, Q1 are collinear, as they appear to be).

Derive and prove a relationship among the distances |SP1|, |SQ1|, and the radius r.

Distance in a tangent figure

Given a circle with center A and radius r and also a point B exterior to the circle. Let line BC be tangent to the circle at B and segment CD be perpendicular to AB. Find and prove a relationship among AB and AD and the radius r.

Another circle relation

Given a circle with center A and a point B exterior to the circle, let E be a point on the circle so that BE=BA and let F be a point on segment AB so that EF=EA. Find and prove a relationship between AB, AF, and the radius of the circle.