What is the length of the altitude drawn to the hypotenuse? The figure is not drawn to scale.
A large right triangle is shown. A perpendicular segment is drawn from the right angle of the large right triangle to the opposite side. That segment divides the opposite side into two parts. The segment on the left is labeled 5. The segment on the right is labeled 14.
(1 point)
Responses
Start Root 70 End Root
Image with alt text: Start Root 70 End Root
70
70
Start Root 19 End Root
Image with alt text: Start Root 19 End Root
19
19
To solve this problem, we can use the Pythagorean theorem.
The two segments on the opposite side of the large right triangle form two smaller right triangles. Let's label the segment of length 5 as a and the segment of length 14 as b.
By the Pythagorean theorem, we have the following equation for the smaller right triangle on the left side:
a^2 + h^2 = 5^2, where h represents the length of the altitude.
Similarly, for the smaller right triangle on the right side, we have:
b^2 + h^2 = 14^2.
Since h^2 appears in both equations, we can subtract the first equation from the second equation to eliminate h^2:
(b^2 + h^2) - (a^2 + h^2) = 14^2 - 5^2.
Simplifying this expression gives us:
b^2 - a^2 = 191.
Now, we can factor the left side of the equation as a difference of squares:
(b - a)(b + a) = 191.
To find the length of the altitude, we need to determine the value of b - a.
From the given information, we know that a = 5 and b = 14. Substituting these values into the equation gives us:
(14 - 5)(14 + 5) = 191.
9 * 19 = 191.
Since 9 * 19 = 171, we can conclude that b - a = 19.
Thus, the length of the altitude drawn to the hypotenuse is 19.
Therefore, the correct answer is:
Start Root 19 End Root.