Questions: Solve the right triangle. Write your answers in simplified, rationalized form. Do UV = m angle V = m angle U =

Transcript text: Solve the right triangle. Write your answers in simplified, rationalized form. Do \[ \begin{aligned} U V & =\square \\ m \angle V & =\square \\ m \angle U & =\square \end{aligned} \]

Solution

Solution Steps

Step 1: Find the length of UV

Given that triangle UVW is a right triangle with UW = VW = 7√22, we can use the Pythagorean theorem to find the length of UV. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (UV) is equal to the sum of the squares of the lengths of the other two sides (UW and VW).

UV² = UW² + VW² UV² = (7√22)² + (7√22)² UV² = 2 * (7√22)² UV² = 2 * 49 * 22 UV² = 2156 UV = √2156 UV = 14√22

Step 2: Find the measure of angle V

Since triangle UVW is a right triangle and UW = VW, this means it is also an isosceles triangle. The angles opposite the equal sides are equal, therefore angles U and V are equal.

Since the sum of angles in a triangle is 180° and angle W = 90°, m∠U + m∠V + m∠W = 180°

Since m∠U = m∠V and m∠W = 90°, we have: 2 * m∠V + 90° = 180° 2 * m∠V = 90° m∠V = 45°

Step 3: Find the measure of angle U

Since m∠U = m∠V, then m∠U = 45°