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# Doubly Isosceles

It is helpful to find what these two triangles have in common. Angle PQR (or SQR), shown in red on the right, is an angle in both triangles.

Triangle PQR is isosceles, so angle PRQ is equal to angle PQR. It is also shown in red in the diagram on the left.

The green angle at P is the angle that needs to be added to two red angles to make 180$^\circ$.

Triangle QRS is also isosceles, and its line of symmetry is through R. So angle RSQ is equal to angle SQR. It is also shown in red in the diagram on the right.

The angle at R also makes a sum of 180$^\circ$ when added to the two red angles. So it is equal to the green angle at P.

But this means that the triangles PQR and QRS are similar.

We can use either that*the scale factor between triangle QRS and triangle PQR is 2* (shown below in the diagram on the left), or that *the ratio between the longer sides and the shorter side of each triangle is 1:2 *(shown below in the diagram on the right).

Either of these facts about similar triangles tell us that QS = $\frac12$.

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Age 14 to 16

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It is helpful to find what these two triangles have in common. Angle PQR (or SQR), shown in red on the right, is an angle in both triangles.

Triangle PQR is isosceles, so angle PRQ is equal to angle PQR. It is also shown in red in the diagram on the left.

The green angle at P is the angle that needs to be added to two red angles to make 180$^\circ$.

Triangle QRS is also isosceles, and its line of symmetry is through R. So angle RSQ is equal to angle SQR. It is also shown in red in the diagram on the right.

The angle at R also makes a sum of 180$^\circ$ when added to the two red angles. So it is equal to the green angle at P.

But this means that the triangles PQR and QRS are similar.

We can use either that

Either of these facts about similar triangles tell us that QS = $\frac12$.

You can find more short problems, arranged by curriculum topic, in our short problems collection.