Question

angela is building the wooden framework for the roof of a house. she needs the angle created by the vertical and horizontal boards of the frame to be a right angle. the height of the vertical board is 12 feet. the length of the horizontal board is 15 feet. the support beam that will connect the ends of the two boards measures 20 feet.
which is true regarding the triangular frame?
it is an acute triangle. about 0.8 foot needs to be removed from the 20 foot board to create a right triangle.
it is an obtuse triangle. about 0.8 foot needs to be removed from the 20 foot board to create a right triangle.
it is an acute triangle. about 7 feet need to be removed from the 20 foot board to create a right triangle.
it is an obtuse triangle. about 7 feet need to be removed from the 20 foot board to create a right triangle.

Explanation:

Step1: Check triangle type (Pythagorean theorem)

For a right triangle, \(a^2 + b^2 = c^2\). Here, \(a = 12\), \(b = 15\), so \(c=\sqrt{12^2 + 15^2}=\sqrt{144 + 225}=\sqrt{369}\approx19.21\). The support beam is 20 feet, which is longer than 19.21. So the triangle with sides 12, 15, 20: check \(12^2 + 15^2 = 144 + 225 = 369\), \(20^2 = 400\). Since \(12^2 + 15^2<20^2\), it's an obtuse triangle (by Pythagorean inequality: \(a^2 + b^2 < c^2\) means obtuse, \(c\) is the longest side).

Step2: Find length to remove

Desired length for right triangle: \(\approx19.21\). Current length: 20. So length to remove: \(20 - 19.21\approx0.8\) feet.

Answer:

It is an obtuse triangle. About 0.8 foot needs to be removed from the 20 foot board to create a right triangle.