Properties of Isosceles Triangles
An isosceles triangle is a triangle that has (at least) two equal side lengths. If all three side lengths are equal, the triangle is also equilateral. Isosceles triangles are very helpful in determining unknown angles.
Terminology
In an isosceles triangle, the two equal sides are called legs, and the remaining side is called the base. The angle opposite the base is called the vertex angle, and the point associated with that angle is called the apex. The two equal angles are called the isosceles angles.
If the triangle is also equilateral, any of the three sides can be considered the base.
Basic Properties
Because angles opposite equal sides are themselves equal, an isosceles triangle has two equal angles (the ones opposite the two equal sides). Thus, given two equal sides and a single angle, the entire structure of the triangle can be determined. Likewise, given two equal angles and the length of any side, the structure of the triangle can be determined.
Determining the area can be done with only a few pieces of information (namely, 3):
- A triangle with base \(b\) and equal side \(\ell\) has area \(\frac{b\sqrt{4\ell^2-b^2}}{4}\).
- A triangle with base \(b\) and isosceles angle \(\theta\) has area \(\frac{b^2\tan \theta}{4}\), as the length of the altitude to the base is \(\frac{b\tan\theta}{2}\). Alternatively, if the apex angle has measure \(\alpha\), the area is \(\frac{b^2}{4}\cot\frac{\alpha}{2}\).
- A triangle with equal side \(\ell\) and isosceles angle \(\theta\) has area \(\frac{\ell^2\sin 2\theta}{2}\), as the vertex angle has measure \(180^{\circ}-2\theta\), and \(\sin(180^{\circ}-2\theta)=\sin 2\theta\). Alternatively, if the vertex angle has measure \(\alpha\), the area is \(\frac{\ell^2\sin\alpha}{2}\).
The altitude to the base also satisfies important properties:
- The altitude to the base is the perpendicular bisector of the base.
- The altitude to the base is the angle bisector of the vertex angle.
- The altitude to the base is the line of symmetry of the triangle.
- The altitude to the base is the median from the apex to the base.
This means that the incenter, circumcenter, centroid, and orthocenter all lie on the altitude to the base, making the altitude to the base the Euler line of the triangle.
Advanced Properties
It is immediate that any \(n\)-sided regular polygon can be decomposed into \(n\) isosceles triangles, where each triangle contains two vertices and the center of the polygon.
\( \begin{align} R &= \frac{S}{2 \sin{\frac{\phi}{2}}} \\ S &= 2 R \sin{\frac{\phi}{2}} \\ r &= R \cos{\frac{\phi}{2}} \\ Area &= \frac{1}{2} R^2 \sin{\phi} \end{align} \)
A regular \(n\)-gon is composed of \(n\) isosceles congruent triangles. Any isosceles triangle is composed of two congruent right triangles as shown in the sketch. These right triangles are very useful in solving \(n\)-gon problems. The relation given could be handy. Apart from the above-mentioned isosceles triangles, there could be many other isoceles triangles in an \(n\)-gon.
\[ n \times \phi =2 \pi = 360^{\circ}. \]
More interestingly, any triangle can be decomposed into \(n\) isosceles triangles, for any positive integer \(n \geq 4\). The picture to the right shows a decomposition of a 13-14-15 triangle into four isosceles triangles.
Problems
Isosceles triangle \(ABC\) has \(AB=AC\) and \(\angle BAC=40^{\circ}\). What is the measure of \(\angle ABC\)?
Because \(AB=AC\), we know that \(\angle ABC=\angle ACB\). Additionally, the sum of the three angles in a triangle is \(180^{\circ}\), so \(\angle ABC+\angle ACB+\angle BAC=2\angle ABC+\angle BAC=180^{\circ}\), and since \(\angle BAC=40^{\circ}\), we have \(2\angle ABC=140^{\circ}\). Thus \(\angle ABC=70^{\circ}\). \(_\square\)
In isosceles triangle \(ABC\), the measure of \(\angle ABC\) is \(50^{\circ}\). What are the possible measures of \(\angle BAC\)?
There are three possible cases:
- Case 1: \(\angle ABC=\angle BAC\). This is easy to deal with, as then \(\angle BAC=50^{\circ}\) immediately.
- Case 2: \(\angle ABC=\angle BCA\). Since the angles of a triangle add up to \(180^{\circ}\), we know that \(50^{\circ}+50^{\circ}+\angle BAC=180^{\circ}\), and so \(\angle BAC=80^{\circ}\).
- Case 3: \(\angle BAC=\angle BCA\). Again, the angles of a triangle add up to \(180^{\circ}\), so \(50^{\circ}+2\angle BAC=180^{\circ}\), and so \(\angle BAC=65^{\circ}\).
Therefore, the possible values of \(\angle BAC\) are \(50^{\circ}, 65^{\circ}\), and \(80^{\circ}\). \(_\square\)
In the above figure, \(AD=DC=CB\) and the measure of \(\angle DAC\) is \(40^{\circ}\). What is the measure of \(\angle DCB\)?
From the given condition, both \(\triangle ADC\) and \(\triangle DCB\) are isosceles triangles.
In \(\triangle ADC\), \(\angle DCA=\angle DAC=40^{\circ}\), implying \[\angle CDB=40^{\circ}+40^{\circ}=80^{\circ}\] by the exterior angle of a triangle.
In \(\triangle DCB\), \(\angle CBD=\angle CDB=80^{\circ}\), implying \[\angle DCB=180^{\circ}-80^{\circ}-80^{\circ}=20^{\circ}\] by the angle sum of a triangle. \(_\square\)
