Draw an inscribed quadrilateral ABCD
and its diagonals intersecting at E
. Name two pairs of similar triangles.
The Correct Answer and Explanation is:
To draw an inscribed quadrilateral ABCD with diagonals intersecting at E, follow these steps and understand the geometric reasoning behind the pairs of similar triangles:
Drawing Instructions
- Draw a circle.
- Mark four points on the circle labeled A, B, C, and D in that order (moving clockwise or counterclockwise).
- Connect the points with straight lines to form the quadrilateral: draw segments AB, BC, CD, and DA.
- Draw the diagonals AC and BD. Let them intersect at point E.
Identifying Similar Triangles
When you draw diagonals AC and BD, they intersect at point E, forming four triangles:
- ΔABE
- ΔCDE
- ΔADE
- ΔCBE
Out of these, we can identify two pairs of similar triangles:
- ΔABE ∼ ΔCDE
- ΔADE ∼ ΔCBE
Explanation
A quadrilateral is said to be cyclic or inscribed when all its vertices lie on the same circle. This means opposite angles of the quadrilateral are supplementary (they add up to 180 degrees). Drawing the diagonals of a cyclic quadrilateral creates two pairs of vertical angles at their point of intersection, which is point E.
Let’s consider the triangles ΔABE and ΔCDE:
- Since both are inscribed in the circle and share the arc properties, angle ABE = angle CDE and angle BAE = angle DCE because they subtend the same arcs.
- The angle at E is vertically opposite, so angle AEB = angle CED.
- With two angles equal, ΔABE ∼ ΔCDE by AA similarity criterion.
Now consider the second pair: ΔADE ∼ ΔCBE
- Again, angle ADE = angle CBE and angle DAE = angle BCE, since they are inscribed angles subtending the same arcs.
- The angles at E are vertically opposite, hence angle DEA = angle CEB.
- Thus, these triangles also satisfy AA similarity.
This geometric relationship is a result of the properties of a circle and how angles subtended by the same arc or chord behave. Understanding these patterns helps identify similarities without needing measurements. These similar triangles are useful in proving other theorems and solving geometry problems involving proportions or angles.
