### All PSAT Math Resources

## Example Questions

### Example Question #21 : Other Polygons

Regular Octagon has sidelength 1.

Give the length of diagonal .

**Possible Answers:**

**Correct answer:**

The trick is to construct segments perpendicular to from and , calling the points of intersection and respectively.

Each interior angle of a regular octagon measures

,

and by symmetry, ,

so .

This makes and triangles.

Since their hypotenuses are sides of the octagon with length 1, then their legs - in particular, and - have length .

Also, since a rectangle was formed when the perpendiculars were drawn, .

The length of diagonal is

.

### Example Question #111 : Plane Geometry

Regular Polygon (a twelve-sided polygon, or *dodecagon*) has sidelength 1.

Give the length of diagonal to the nearest tenth.

**Possible Answers:**

**Correct answer:**

The trick is to construct segments perpendicular to from and , calling the points of intersection and respectively.

Each interior angle of a regular dodecagon measures

.

Since and are perpendicular to , it can be shown via symmetry that they are also perpendicular to . Therefore,

and both measure

and and are triangles with long legs and . Since their hypotenuses are sides of the dodecagon and therefore have length 1,

.

Also, since Quadrilateral is a rectangle, .

The length of diagonal is.

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