Let F be the midpoint of side BC of triangle ABC. Construct isosceles right triangles ABD and ACE externally on sides AB and AC with the right angles at D and E, respectively. What type of triangle is DEF.
right angled isosceles
To determine the type of triangle DEF, we need to consider the angles of triangle ABC and the construction of triangle DEF.
1. Given the construction, triangle ABC is a scalene triangle since none of its sides are equal, and it has three different angles.
2. Triangle ABD is an isosceles right triangle, meaning it has a right angle at D and two equal angles at A and B. This implies that angle ABD = angle BAD, and AB = BD.
3. Similarly, triangle ACE is an isosceles right triangle, where angle CAE = angle EAC, and AC = CE.
4. Since F is the midpoint of side BC, we can conclude that BF = FC.
Now, let's consider triangle DEF:
5. Angle ADE = 90 degrees (as triangle ABD is a right triangle).
6. Angle ADF is formed by two adjacent angles: angle ADE and angle EDF. Since angle ADE = 90 degrees, we can conclude that angle ADF + angle EDF = 90 degrees.
7. Similarly, angle AEF = 90 degrees (as triangle ACE is a right triangle).
8. Since angle ADF + angle EDF = 90 degrees, and angle AEF = 90 degrees, we can conclude that angle ADF + angle EDF + angle AEF = 90 degrees.
9. From step 4, we know that BF = FC. Since triangle ABC is scalene, angle B and angle C are unequal. This means that angles ABC and ACB are unequal as well.
10. Triangle DFE is formed by connecting points D, F, and E. Since angle ADF + angle EDF + angle AEF = 90 degrees (as established in step 8), and angles ABC and ACB are unequal, we can conclude that triangle DEF is a scalene triangle.
Therefore, triangle DEF is a scalene triangle.
To determine the type of triangle DEF, we first need to construct triangles ABD and ACE externally on sides AB and AC.
1. Begin by drawing triangle ABC with side lengths AB, BC, and AC.
2. Locate the midpoint, F, on side BC. Draw a line segment from point F to point A to divide triangle ABC into two smaller triangles, namely, AFB and AFC.
3. Now, construct isosceles right triangle ABD externally on side AB. To do this, extend side AB past point B and construct a right triangle ABD with right angle at D, such that AD is equal to BD.
4. Next, construct isosceles right triangle ACE externally on side AC. Extend side AC past point C, and construct a right triangle ACE with right angle at E, such that AE is equal to CE.
After following the steps above, triangle DEF is formed by connecting the points D, E, and F.
Based on the construction, we can observe the following properties:
1. Triangle ABD is an isosceles right triangle, meaning that it has two congruent sides (AB and AD) and a right angle (at point D).
2. Triangle ACE is also an isosceles right triangle, having two congruent sides (AC and AE) and a right angle (at point E).
From the construction, we can deduce that:
3. Triangle DEF is an isosceles triangle, as it is formed by connecting the midpoints (F) of two congruent sides (BD and CE) of two isosceles triangles (ABD and ACE).
4. Since both ABD and ACE are right triangles, triangle DEF will have each of its angles measure 90 degrees. Therefore, triangle DEF is also a right triangle.
In conclusion, triangle DEF is an isosceles right triangle.