Seth is using the figure shown below to prove the pythagorean theorem using triangle similarity:

in the given triangle def, angle d is 90° and segment dg is perpendicular to segment ef.

the figure shows triangle def with right angle at d and segment dg. point g is on side ef.

part a: identify a pair of similar triangles. (2 points)

part b: explain how you know the triangles from part a are similar. (4 points)

part c: if eg = 2 and ef = 8, find the length of segment ed. show your work. (4 points)

The similar triangles are triangle DEG and triangle DFG

Segment ED = 2.83 units

Step-by-step explanation:

In the triangle DEF, note that angle D is 90°, that is, its a right angle. Therefore if triangle DEF is right angled then the sides can be calculated using the Pythagoras theorem. Also, a line is drawn from point D and is perpendicular to line EF. This means line DG divides line EF into two equal 90 degree angles (angles on a straight line equals 180°). Having been given that angle D is a 90 degree angle and the line that connects it to line EF is perpendicular, then line DG has divided angle D into two equal halves and that means each half is 45°.

Therefore we have two similar triangles and they are labelled as triangle DEG and triangle DFG.

Triangle DEG has angle G measuring 90 degrees, angle D measuring 45 degrees and angle E derived as

E = 180 - (90 + 45) [sum of angles in a triangle equals 180]

E = 180 - 135

E = 45

Also in triangle DFG, you have angle G measuring 90 degrees, angle D measuring 45 degrees and angle F derived as

F = 180 - (90 + 45)

F = 180 - 135

F = 45

Therefore triangles DEG and DFG are similar based on AAA parameters.

Part B;

To calculate the length of segment ED, we start by isolating one out of the two similar triangles and that would be triangle DEG.

Triangle DEG is a right isosceles triangle. We know this because two angles have been identified as 45 degrees each. In an isosceles triangle, two angles are equal and the two sides corresponding to the two equal angles are also equal. Line EG is opposite angle D, in the same way line DG is opposite angle E

If angle D = angle E (45 degrees)

Then line EG = line DG (2 units)

Hence, in triangle DEG, line ED is calculated using the Pythagoras theorem as follows;

ED² = EG² + DG²

Where ED is the hypotenuse and EG and DG are the two other legs/sides

ED² = 2² + 2²

ED² = 4 + 4

ED² = 8

Add the square root sign to both sides of the equation

√ED² = √8

ED = 2.8284

ED ≈ 2.83