Points E, D, and H are the midpoints of the sides of TUV.UV=100, TV=120, and HD=100.

Answer 1

Since E is the midpoint of TU, we have ET = EU = 120/2 = 60.
Similarly, we have TD = TH = 100/2 = 50.
Since HD = 100, we have DH = 100/2 = 50.
Therefore, DHT is an isosceles triangle with DH = TH = 50.
Since the sum of the measures of the angles in a triangle is 180 degrees, we have $\angle DHT = \frac{180 - \angle TDH}{2} = \frac{180 - \angle HTD}{2}$.
Adding these two equations, we get $2 \angle DHT = \frac{180 - \angle HTD}{2} + \frac{180 - \angle TDH}{2}$.
Simplifying the right side of the equation, we get $2 \angle DHT = \frac{360 - \angle HTD - \angle HTD}{2} = \frac{360 - \angle HTD}{2}$.
Multiplying both sides of the equation by 2 gives us $4 \angle DHT = 360 - \angle HTD$, so $\angle DHT + \angle HTD = 360/4 = 90$.
Therefore, $\angle DHT = \angle HDT = 45$.
Since we have a right angle at DHT, the sum of the angles is 180 degrees, so $\angle HTD = 90 - \angle DHT - \angle HDT = 180 - 2 \cdot 45 = 90$.
Since $\angle HTD = 90$ and $\angle DHT = \angle HDT = 45$, we have a right isosceles triangle with HT = HD = 50.