Question

solve for the other leg and for the hypotenuse of the 45-45-90 triangle
(image of a right triangle with one leg labeled 7, one angle labeled 45°, hypotenuse labeled u, and the other leg labeled v)
options:
a. u=7 and v=7
b. u = 7 and v = 7√2
c. u = 7√2 and v = 7
d. u = 7√2 and v = 7√2

Explanation:

Step1: Analyze the 45-45-90 triangle properties

In a 45-45-90 triangle, the two legs are equal in length, and the hypotenuse \( h \) is related to the leg length \( l \) by \( h = l\sqrt{2} \). Here, one leg is 7, so the other leg \( v \) (since it's a 45-45-90 triangle, the legs are equal) should also be 7. Wait, no, wait: Wait, the angle given is 45, and one leg is 7. Wait, actually, in a 45-45-90 triangle, the legs are congruent, so if one leg is 7, the other leg \( v \) is 7. Then the hypotenuse \( u \) is \( 7\sqrt{2} \)? Wait, no, wait the labels: the side labeled 7 is a leg, the right angle is between the leg of length 7 and \( v \), and the hypotenuse is \( u \). Wait, no, let's re-examine. Wait, the triangle has a right angle, one angle 45, so the other angle is 45 (since angles in a triangle sum to 180: 90 + 45 + 45 = 180). So it's an isosceles right triangle, so the two legs are equal. So the leg with length 7 and the leg \( v \) are equal, so \( v = 7 \). Then the hypotenuse \( u \) is leg \( \times \sqrt{2} \), so \( u = 7\sqrt{2} \). Wait, but looking at the options: option c is \( u = 7\sqrt{2} \) and \( v = 7 \). Let's check the options again.

Wait, maybe I mixed up the labels. Let's see the diagram: the vertical leg is 7, the horizontal leg is \( v \), and the hypotenuse is \( u \). Since it's a 45-45-90 triangle, the two legs are equal, so \( v = 7 \). Then the hypotenuse \( u = 7\sqrt{2} \). So that's option c: \( u = 7\sqrt{2} \) and \( v = 7 \).

Wait, but let's confirm the properties again. In a 45-45-90 triangle, legs are equal, hypotenuse is leg sqrt(2). So if one leg is 7, the other leg (v) is 7, hypotenuse (u) is 7sqrt(2). Wait, no, wait the options: option c is u=7√2 and v=7. So that's correct.

Step2: Match with the options

Given the properties, the other leg (v) is equal to 7 (since legs are equal in 45-45-90), and the hypotenuse (u) is 7√2. So the correct option is c.

Answer:

c. \( u = 7\sqrt{2} \) and \( v = 7 \)