### At a Glance

The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?

### No Right Angle Here

Prove that the internal angle bisectors of a triangle will never be perpendicular to each other.

### A Sameness Surely

Triangle ABC has a right angle at C. ACRS and CBPQ are squares. ST and PU are perpendicular to AB produced. Show that ST + PU = AB

# Triangle Split

##### Stage: 3 and 4 Short Challenge Level:

Since $SP = SQ$, the triangle $PSQ$ is isosceles. Therefore, $\angle SPQ = \angle PQS$. We denote the measure of those angles by $y$. Similarly, $\angle SQR = \angle QRS = x^\circ$.

Since the sums of the interior angles of $PQR$ is $180^\circ$, $x + y + (x+y) = 180$, so $2x+2y=180$.

Therefore, $\angle PQR = x^\circ+y^\circ=90^\circ$.

This problem is taken from the UKMT Mathematical Challenges.
View the archive of all weekly problems grouped by curriculum topic

View the previous week's solution
View the current weekly problem