We can assume that these two triangles are similar (meaning that they’re not congruent, but proportional) because vertical angles are opposite and have congruent measures due to the intersection created. They also have a right angle (90 degrees) in common. Therefore they are similar under the AA-similarity postulate.
Note: the “m” in front means of a number “the measure of.”
First compare corresponding sides:
the base of the top triangle (m3) corresponds to the base of the bottom triangle (m3.5)
the hypotenuse of the top triangle (x) corresponds to the hypotenuse of the bottom (m7).
Based on these measurements the bottom triangle is larger than the top.
Set up the equation, method no.1:
3/3.5 = x/7 (cross multiply)
(3 × 7) = (3.5 × x)
21 = 3.5x
21/3.5x = 3.5x/3.5 (division prop. of equality)
6 = x
OR, method no. 2:
3/3.5 = 0.857142…
0.857142 × 7 = 5.99999… (which is equivalent to 6)
x = 6
If we were to find the measure of the first legs (3rd and remaining side) for either triangle, they’d be proportional too.
Notice:
c² = a² + b²
7² = 3.5² + b²
49 = 12.25 + b²
49 - 12.25 = b² - 12.25
36.75 = b²
√36.75 = √b²
6.062… = b
3/3.5 = y (as in the top triangle’s leg)/6.062
y = 5.196…
5.196 : 6.062 also equals 0.857142… so by SSS Postulate all side and angles are proportional.
Triangles where the right angles (or vertex angles) are meet. View for comparison and proportionality:
